(*

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Copyright (C) 2005 - 2008 Slawomir Kolodynski

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header{*\isaheader{Topology\_ZF\_2.thy}*}

theory Topology_ZF_2 imports Topology_ZF_1 func1 Fol1

begin

text{*This theory continues the series on general topology and covers the

definition and basic properties of continuous functions. We also introduce the notion of

homeomorphism an prove the pasting lemma. *}

section{*Continuous functions.*}

text{*In this section we define continuous functions and prove that certain

conditions are equivalent to a function being continuous.*}

text{*In standard math we say that a function is contiuous with respect to two

topologies $\tau_1 ,\tau_2 $ if the inverse image of sets from topology

$\tau_2$ are in $\tau_1$. Here we define a predicate that is supposed

to reflect that definition, with a difference that we don't require in the

definition that $\tau_1 ,\tau_2 $ are topologies. This means for example that

when we define measurable functions, the definition will be the same.

The notation @{text "f-``(A)"} means the inverse image of (a set)

$A$ with respect to (a function) $f$.

*}

definition

"IsContinuous(τ⇩_{1},τ⇩_{2},f) ≡ (∀U∈τ⇩_{2}. f-``(U) ∈ τ⇩_{1})"

text{*A trivial example of a continuous function - identity is continuous.*}

lemma id_cont: shows "IsContinuous(τ,τ,id(\<Union>τ))"

proof -

{ fix U assume "U∈τ"

then have "id(\<Union>τ)-``(U) = U" using vimage_id_same by auto

with `U∈τ` have "id(\<Union>τ)-``(U) ∈ τ" by simp

} then show "IsContinuous(τ,τ,id(\<Union>τ))" using IsContinuous_def

by simp

qed

text{*We will work with a pair of topological spaces. The following

locale sets up our context that consists of

two topologies $\tau_1,\tau_2$ and

a continuous function $f: X_1 \rightarrow X_2$, where $X_i$ is defined

as $\bigcup\tau_i$ for $i=1,2$. We also define notation @{text "cl⇩_{1}(A)"} and

@{text "cl⇩_{2}(A)"} for closure of a set $A$ in topologies $\tau_1$ and $\tau_2$,

respectively.*}

locale two_top_spaces0 =

fixes τ⇩_{1}

assumes tau1_is_top: "τ⇩_{1}{is a topology}"

fixes τ⇩_{2}

assumes tau2_is_top: "τ⇩_{2}{is a topology}"

fixes X⇩_{1}

defines X1_def [simp]: "X⇩_{1}≡ \<Union>τ⇩_{1}"

fixes X⇩_{2}

defines X2_def [simp]: "X⇩_{2}≡ \<Union>τ⇩_{2}"

fixes f

assumes fmapAssum: "f: X⇩_{1}-> X⇩_{2}"

fixes isContinuous ("_ {is continuous}" [50] 50)

defines isContinuous_def [simp]: "g {is continuous} ≡ IsContinuous(τ⇩_{1},τ⇩_{2},g)"

fixes cl⇩_{1}

defines cl1_def [simp]: "cl⇩_{1}(A) ≡ Closure(A,τ⇩_{1})"

fixes cl⇩_{2}

defines cl2_def [simp]: "cl⇩_{2}(A) ≡ Closure(A,τ⇩_{2})"

text{*First we show that theorems proven in locale @{text "topology0"}

are valid when applied to topologies $\tau_1$ and $\tau_2$.*}

lemma (in two_top_spaces0) topol_cntxs_valid:

shows "topology0(τ⇩_{1})" and "topology0(τ⇩_{2})"

using tau1_is_top tau2_is_top topology0_def by auto

text{*For continuous functions the inverse image of a closed set is closed.*}

lemma (in two_top_spaces0) TopZF_2_1_L1:

assumes A1: "f {is continuous}" and A2: "D {is closed in} τ⇩_{2}"

shows "f-``(D) {is closed in} τ⇩_{1}"

proof -

from fmapAssum have "f-``(D) ⊆ X⇩_{1}" using func1_1_L3 by simp

moreover from fmapAssum have "f-``(X⇩_{2}- D) = X⇩_{1}- f-``(D)"

using Pi_iff function_vimage_Diff func1_1_L4 by auto

ultimately have "X⇩_{1}- f-``(X⇩_{2}- D) = f-``(D)" by auto

moreover from A1 A2 have "(X⇩_{1}- f-``(X⇩_{2}- D)) {is closed in} τ⇩_{1}"

using IsClosed_def IsContinuous_def topol_cntxs_valid topology0.Top_3_L9

by simp

ultimately show "f-``(D) {is closed in} τ⇩_{1}" by simp

qed

text{*If the inverse image of every closed set is closed, then the

image of a closure is contained in the closure of the image.*}

lemma (in two_top_spaces0) Top_ZF_2_1_L2:

assumes A1: "∀D. ((D {is closed in} τ⇩_{2}) --> f-``(D) {is closed in} τ⇩_{1})"

and A2: "A ⊆ X⇩_{1}"

shows "f``(cl⇩_{1}(A)) ⊆ cl⇩_{2}(f``(A))"

proof -

from fmapAssum have "f``(A) ⊆ cl⇩_{2}(f``(A))"

using func1_1_L6 topol_cntxs_valid topology0.cl_contains_set

by simp

with fmapAssum have "f-``(f``(A)) ⊆ f-``(cl⇩_{2}(f``(A)))"

by auto;

moreover from fmapAssum A2 have "A ⊆ f-``(f``(A))"

using func1_1_L9 by simp

ultimately have "A ⊆ f-``(cl⇩_{2}(f``(A)))" by auto

with fmapAssum A1 have "f``(cl⇩_{1}(A)) ⊆ f``(f-``(cl⇩_{2}(f``(A))))"

using func1_1_L6 func1_1_L8 IsClosed_def

topol_cntxs_valid topology0.cl_is_closed topology0.Top_3_L13

by simp

moreover from fmapAssum have "f``(f-``(cl⇩_{2}(f``(A)))) ⊆ cl⇩_{2}(f``(A))"

using fun_is_function function_image_vimage by simp

ultimately show "f``(cl⇩_{1}(A)) ⊆ cl⇩_{2}(f``(A))"

by auto

qed

text{*If $f\left( \overline{A}\right)\subseteq \overline{f(A)}$

(the image of the closure is contained in the closure of the image), then

$\overline{f^{-1}(B)}\subseteq f^{-1}\left( \overline{B} \right)$

(the inverse image of the closure contains the closure of the

inverse image).*}

lemma (in two_top_spaces0) Top_ZF_2_1_L3:

assumes A1: "∀ A. ( A ⊆ X⇩_{1}--> f``(cl⇩_{1}(A)) ⊆ cl⇩_{2}(f``(A)))"

shows "∀B. ( B ⊆ X⇩_{2}--> cl⇩_{1}(f-``(B)) ⊆ f-``(cl⇩_{2}(B)) )"

proof -

{ fix B assume "B ⊆ X⇩_{2}"

from fmapAssum A1 have "f``(cl⇩_{1}(f-``(B))) ⊆ cl⇩_{2}(f``(f-``(B)))"

using func1_1_L3 by simp

moreover from fmapAssum `B ⊆ X⇩_{2}` have "cl⇩_{2}(f``(f-``(B))) ⊆ cl⇩_{2}(B)"

using fun_is_function function_image_vimage func1_1_L6

topol_cntxs_valid topology0.top_closure_mono

by simp

ultimately have "f-``(f``(cl⇩_{1}(f-``(B)))) ⊆ f-``(cl⇩_{2}(B))"

using fmapAssum fun_is_function by auto;

moreover from fmapAssum `B ⊆ X⇩_{2}` have

"cl⇩_{1}(f-``(B)) ⊆ f-``(f``(cl⇩_{1}(f-``(B))))"

using func1_1_L3 func1_1_L9 IsClosed_def

topol_cntxs_valid topology0.cl_is_closed by simp

ultimately have "cl⇩_{1}(f-``(B)) ⊆ f-``(cl⇩_{2}(B))" by auto

} then show ?thesis by simp

qed;

text{*If $\overline{f^{-1}(B)}\subseteq f^{-1}\left( \overline{B} \right)$

(the inverse image of a closure contains the closure of the

inverse image), then the function is continuous. This lemma closes a series of

implications in lemmas @{text " Top_ZF_2_1_L1"},

@{text " Top_ZF_2_1_L2"} and @{text " Top_ZF_2_1_L3"} showing equivalence

of four definitions of continuity.*}

lemma (in two_top_spaces0) Top_ZF_2_1_L4:

assumes A1: "∀B. ( B ⊆ X⇩_{2}--> cl⇩_{1}(f-``(B)) ⊆ f-``(cl⇩_{2}(B)) )"

shows "f {is continuous}"

proof -

{ fix U assume "U ∈ τ⇩_{2}"

then have "(X⇩_{2}- U) {is closed in} τ⇩_{2}"

using topol_cntxs_valid topology0.Top_3_L9 by simp;

moreover have "X⇩_{2}- U ⊆ \<Union>τ⇩_{2}" by auto

ultimately have "cl⇩_{2}(X⇩_{2}- U) = X⇩_{2}- U"

using topol_cntxs_valid topology0.Top_3_L8 by simp

moreover from A1 have "cl⇩_{1}(f-``(X⇩_{2}- U)) ⊆ f-``(cl⇩_{2}(X⇩_{2}- U))"

by auto

ultimately have "cl⇩_{1}(f-``(X⇩_{2}- U)) ⊆ f-``(X⇩_{2}- U)" by simp

moreover from fmapAssum have "f-``(X⇩_{2}- U) ⊆ cl⇩_{1}(f-``(X⇩_{2}- U))"

using func1_1_L3 topol_cntxs_valid topology0.cl_contains_set

by simp

ultimately have "f-``(X⇩_{2}- U) {is closed in} τ⇩_{1}"

using fmapAssum func1_1_L3 topol_cntxs_valid topology0.Top_3_L8

by auto

with fmapAssum have "f-``(U) ∈ τ⇩_{1}"

using fun_is_function function_vimage_Diff func1_1_L4

func1_1_L3 IsClosed_def double_complement by simp

} then have "∀U∈τ⇩_{2}. f-``(U) ∈ τ⇩_{1}" by simp

then show ?thesis using IsContinuous_def by simp

qed;

text{*Another condition for continuity: it is sufficient to check if the

inverse image of every set in a base is open.*}

lemma (in two_top_spaces0) Top_ZF_2_1_L5:

assumes A1: "B {is a base for} τ⇩_{2}" and A2: "∀U∈B. f-``(U) ∈ τ⇩_{1}"

shows "f {is continuous}"

proof -

{ fix V assume A3: "V ∈ τ⇩_{2}"

with A1 obtain A where "A ⊆ B" "V = \<Union>A"

using IsAbaseFor_def by auto

with A2 have "{f-``(U). U∈A} ⊆ τ⇩_{1}" by auto

with tau1_is_top have "\<Union> {f-``(U). U∈A} ∈ τ⇩_{1}"

using IsATopology_def by simp

moreover from `A ⊆ B` `V = \<Union>A` have "f-``(V) = \<Union>{f-``(U). U∈A}"

by auto;

ultimately have "f-``(V) ∈ τ⇩_{1}" by simp

} then show "f {is continuous}" using IsContinuous_def

by simp

qed;

text{*We can strenghten the previous lemma: it is sufficient to check if the

inverse image of every set in a subbase is open. The proof is rather awkward,

as usual when we deal with general intersections. We have to keep track of

the case when the collection is empty.*}

lemma (in two_top_spaces0) Top_ZF_2_1_L6:

assumes A1: "B {is a subbase for} τ⇩_{2}" and A2: "∀U∈B. f-``(U) ∈ τ⇩_{1}"

shows "f {is continuous}"

proof -

let ?C = "{\<Inter>A. A ∈ FinPow(B)}"

from A1 have "?C {is a base for} τ⇩_{2}"

using IsAsubBaseFor_def by simp

moreover have "∀U∈?C. f-``(U) ∈ τ⇩_{1}"

proof

fix U assume "U∈?C"

{ assume "f-``(U) = 0"

with tau1_is_top have "f-``(U) ∈ τ⇩_{1}"

using empty_open by simp }

moreover

{ assume "f-``(U) ≠ 0"

then have "U≠0" by (rule func1_1_L13)

moreover from `U∈?C` obtain A where

"A ∈ FinPow(B)" and "U = \<Inter>A"

by auto

ultimately have "\<Inter>A≠0" by simp

then have "A≠0" by (rule inter_nempty_nempty)

then have "{f-``(W). W∈A} ≠ 0" by simp

moreover from A2 `A ∈ FinPow(B)` have "{f-``(W). W∈A} ∈ FinPow(τ⇩_{1})"

by (rule fin_image_fin)

ultimately have "\<Inter>{f-``(W). W∈A} ∈ τ⇩_{1}"

using topol_cntxs_valid topology0.fin_inter_open_open by simp

moreover

from `A ∈ FinPow(B)` have "A ⊆ B" using FinPow_def by simp

with tau2_is_top A1 have "A ⊆ Pow(X⇩_{2})"

using IsAsubBaseFor_def IsATopology_def by auto

with fmapAssum `A≠0` `U = \<Inter>A` have "f-``(U) = \<Inter>{f-``(W). W∈A}"

using func1_1_L12 by simp

ultimately have "f-``(U) ∈ τ⇩_{1}" by simp }

ultimately show "f-``(U) ∈ τ⇩_{1}" by blast

qed

ultimately show "f {is continuous}"

using Top_ZF_2_1_L5 by simp

qed

text{*A dual of @{text " Top_ZF_2_1_L5"}: a function that maps base sets to open sets

is open.*}

lemma (in two_top_spaces0) base_image_open:

assumes A1: "\<B> {is a base for} τ⇩_{1}" and A2: "∀B∈\<B>. f``(B) ∈ τ⇩_{2}" and A3: "U∈τ⇩_{1}"

shows "f``(U) ∈ τ⇩_{2}"

proof -

from A1 A3 obtain \<E> where "\<E> ∈ Pow(\<B>)" and "U = \<Union>\<E>" using Top_1_2_L1 by blast

with A1 have "f``(U) = \<Union>{f``(E). E ∈ \<E>}" using Top_1_2_L5 fmapAssum image_of_Union

by auto

moreover

from A2 `\<E> ∈ Pow(\<B>)` have "{f``(E). E ∈ \<E>} ∈ Pow(τ⇩_{2})" by auto

then have "\<Union>{f``(E). E ∈ \<E>} ∈ τ⇩_{2}" using tau2_is_top IsATopology_def by simp

ultimately show ?thesis using tau2_is_top IsATopology_def by auto

qed

text{*A composition of two continuous functions is continuous.*}

lemma comp_cont: assumes "IsContinuous(T,S,f)" and "IsContinuous(S,R,g)"

shows "IsContinuous(T,R,g O f)"

using assms IsContinuous_def vimage_comp by simp

text{*A composition of three continuous functions is continuous.*}

lemma comp_cont3:

assumes "IsContinuous(T,S,f)" and "IsContinuous(S,R,g)" and "IsContinuous(R,P,h)"

shows "IsContinuous(T,P,h O g O f)"

using assms IsContinuous_def vimage_comp by simp

section{*Homeomorphisms*}

text{*This section studies ''homeomorphisms'' - continous bijections whose inverses

are also continuous. Notions that are preserved by (commute with)

homeomorphisms are called ''topological invariants''. *}

text{*Homeomorphism is a bijection that preserves open sets.*}

definition "IsAhomeomorphism(T,S,f) ≡

f ∈ bij(\<Union>T,\<Union>S) ∧ IsContinuous(T,S,f) ∧ IsContinuous(S,T,converse(f))"

text{*Inverse (converse) of a homeomorphism is a homeomorphism.*}

lemma homeo_inv: assumes "IsAhomeomorphism(T,S,f)"

shows "IsAhomeomorphism(S,T,converse(f))"

using assms IsAhomeomorphism_def bij_converse_bij bij_converse_converse

by auto

text{*Homeomorphisms are open maps.*}

lemma homeo_open: assumes "IsAhomeomorphism(T,S,f)" and "U∈T"

shows "f``(U) ∈ S"

using assms image_converse IsAhomeomorphism_def IsContinuous_def by simp

text{*A continuous bijection that is an open map is a homeomorphism.*}

lemma bij_cont_open_homeo:

assumes "f ∈ bij(\<Union>T,\<Union>S)" and "IsContinuous(T,S,f)" and "∀U∈T. f``(U) ∈ S"

shows "IsAhomeomorphism(T,S,f)"

using assms image_converse IsAhomeomorphism_def IsContinuous_def by auto

text{*A continuous bijection that maps base to open sets is a homeomorphism.*}

lemma (in two_top_spaces0) bij_base_open_homeo:

assumes A1: "f ∈ bij(X⇩_{1},X⇩_{2})" and A2: "\<B> {is a base for} τ⇩_{1}" and A3: "\<C> {is a base for} τ⇩_{2}" and

A4: "∀U∈\<C>. f-``(U) ∈ τ⇩_{1}" and A5: "∀V∈\<B>. f``(V) ∈ τ⇩_{2}"

shows "IsAhomeomorphism(τ⇩_{1},τ⇩_{2},f)"

using assms tau2_is_top tau1_is_top bij_converse_bij bij_is_fun two_top_spaces0_def

image_converse two_top_spaces0.Top_ZF_2_1_L5 IsAhomeomorphism_def by simp

text{*A bijection that maps base to base is a homeomorphism.*}

lemma (in two_top_spaces0) bij_base_homeo:

assumes A1: "f ∈ bij(X⇩_{1},X⇩_{2})" and A2: "\<B> {is a base for} τ⇩_{1}" and

A3: "{f``(B). B∈\<B>} {is a base for} τ⇩_{2}"

shows "IsAhomeomorphism(τ⇩_{1},τ⇩_{2},f)"

proof -

note A1

moreover have "f {is continuous}"

proof -

{ fix C assume "C ∈ {f``(B). B∈\<B>}"

then obtain B where "B∈\<B>" and I: "C = f``(B)" by auto

with A2 have "B ⊆ X⇩_{1}" using Top_1_2_L5 by auto

with A1 A2 `B∈\<B>` I have "f-``(C) ∈ τ⇩_{1}"

using bij_def inj_vimage_image base_sets_open by auto

} hence "∀C ∈ {f``(B). B∈\<B>}. f-``(C) ∈ τ⇩_{1}" by auto

with A3 show ?thesis by (rule Top_ZF_2_1_L5)

qed

moreover

from A3 have "∀B∈\<B>. f``(B) ∈ τ⇩_{2}" using base_sets_open by auto

with A2 have "∀U∈τ⇩_{1}. f``(U) ∈ τ⇩_{2}" using base_image_open by simp

ultimately show ?thesis using bij_cont_open_homeo by simp

qed

text{*Interior is a topological invariant.*}

theorem int_top_invariant: assumes A1: "A⊆\<Union>T" and A2: "IsAhomeomorphism(T,S,f)"

shows "f``(Interior(A,T)) = Interior(f``(A),S)"

proof -

let ?\<A> = "{U∈T. U⊆A}"

have I: "{f``(U). U∈?\<A>} = {V∈S. V ⊆ f``(A)}"

proof

from A2 show "{f``(U). U∈?\<A>} ⊆ {V∈S. V ⊆ f``(A)}"

using homeo_open by auto

{ fix V assume "V ∈ {V∈S. V ⊆ f``(A)}"

hence "V∈S" and II: "V ⊆ f``(A)" by auto

let ?U = "f-``(V)"

from II have "?U ⊆ f-``(f``(A))" by auto

moreover from assms have "f-``(f``(A)) = A"

using IsAhomeomorphism_def bij_def inj_vimage_image by auto

moreover from A2 `V∈S` have "?U∈T"

using IsAhomeomorphism_def IsContinuous_def by simp

moreover

from `V∈S` have "V ⊆ \<Union>S" by auto

with A2 have "V = f``(?U)"

using IsAhomeomorphism_def bij_def surj_image_vimage by auto

ultimately have "V ∈ {f``(U). U∈?\<A>}" by auto

} thus "{V∈S. V ⊆ f``(A)} ⊆ {f``(U). U∈?\<A>}" by auto

qed

have "f``(Interior(A,T)) = f``(\<Union>?\<A>)" unfolding Interior_def by simp

also from A2 have "… = \<Union>{f``(U). U∈?\<A>}"

using IsAhomeomorphism_def bij_def inj_def image_of_Union by auto

also from I have "… = Interior(f``(A),S)" unfolding Interior_def by simp

finally show ?thesis by simp

qed

section{*Topologies induced by mappings*}

text{*In this section we consider various ways a topology may be defined on a set that

is the range (or the domain) of a function whose domain (or range) is a topological space.

*}

text{*A bijection from a topological space induces a topology on the range.*}

theorem bij_induced_top: assumes A1: "T {is a topology}" and A2: "f ∈ bij(\<Union>T,Y)"

shows

"{f``(U). U∈T} {is a topology}" and

"{ {f`(x).x∈U}. U∈T} {is a topology}" and

"(\<Union>{f``(U). U∈T}) = Y" and

"IsAhomeomorphism(T, {f``(U). U∈T},f)"

proof -

from A2 have "f ∈ inj(\<Union>T,Y)" using bij_def by simp

then have "f:\<Union>T->Y" using inj_def by simp

let ?S = "{f``(U). U∈T}"

{ fix M assume "M ∈ Pow(?S)"

let ?M⇩_{T}= "{f-``(V). V∈M}"

have "?M⇩_{T}⊆ T"

proof

fix W assume "W∈?M⇩_{T}"

then obtain V where "V∈M" and I: "W = f-``(V)" by auto

with `M ∈ Pow(?S)` have "V∈?S" by auto

then obtain U where "U∈T" and "V = f``(U)" by auto

with I have "W = f-``(f``(U))" by simp

with `f ∈ inj(\<Union>T,Y)` `U∈T` have "W = U" using inj_vimage_image by blast

with `U∈T` show "W∈T" by simp

qed

with A1 have "(\<Union>?M⇩_{T}) ∈ T" using IsATopology_def by simp

hence "f``(\<Union>?M⇩_{T}) ∈ ?S" by auto

moreover have "f``(\<Union>?M⇩_{T}) = \<Union>M"

proof -

from `f:\<Union>T->Y` `?M⇩_{T}⊆ T` have "f``(\<Union>?M⇩_{T}) = \<Union>{f``(U). U∈?M⇩_{T}}"

using image_of_Union by auto

moreover have "{f``(U). U∈?M⇩_{T}} = M"

proof -

from `f:\<Union>T->Y` have "∀U∈T. f``(U) ⊆ Y" using func1_1_L6 by simp

with `M ∈ Pow(?S)` have "M ⊆ Pow(Y)" by auto

with A2 show "{f``(U). U∈?M⇩_{T}} = M" using bij_def surj_subsets by auto

qed

ultimately show "f``(\<Union>?M⇩_{T}) = \<Union>M" by simp

qed

ultimately have "\<Union>M ∈ ?S" by auto

} then have "∀M∈Pow(?S). \<Union>M ∈ ?S" by auto

moreover

{ fix U V assume "U∈?S" "V∈?S"

then obtain U⇩_{T}V⇩_{T}where "U⇩_{T}∈ T" "V⇩_{T}∈ T" and

I: "U = f``(U⇩_{T})" "V = f``(V⇩_{T})"

by auto

with A1 have "U⇩_{T}∩V⇩_{T}∈ T" using IsATopology_def by simp

hence "f``(U⇩_{T}∩V⇩_{T}) ∈ ?S" by auto

moreover have "f``(U⇩_{T}∩V⇩_{T}) = U∩V"

proof -

from `U⇩_{T}∈ T` `V⇩_{T}∈ T` have "U⇩_{T}⊆ \<Union>T" "V⇩_{T}⊆ \<Union>T"

using bij_def by auto

with `f ∈ inj(\<Union>T,Y)` I show "f``(U⇩_{T}∩V⇩_{T}) = U∩V" using inj_image_inter

by simp

qed

ultimately have "U∩V ∈ ?S" by simp

} then have "∀U∈?S. ∀V∈?S. U∩V ∈ ?S" by auto

ultimately show "?S {is a topology}" using IsATopology_def by simp

moreover from `f:\<Union>T->Y` have "∀U∈T. f``(U) = {f`(x).x∈U}"

using func_imagedef by blast

ultimately show "{ {f`(x).x∈U}. U∈T} {is a topology}" by simp

show "\<Union>?S = Y"

proof

from `f:\<Union>T->Y` have "∀U∈T. f``(U) ⊆ Y" using func1_1_L6 by simp

thus "\<Union>?S ⊆ Y" by auto

from A1 have "f``(\<Union>T) ⊆ \<Union>?S" using IsATopology_def by auto

with A2 show "Y ⊆ \<Union>?S" using bij_def surj_range_image_domain

by auto

qed

show "IsAhomeomorphism(T,?S,f)"

proof -

from A2 `\<Union>?S = Y` have "f ∈ bij(\<Union>T,\<Union>?S)" by simp

moreover have "IsContinuous(T,?S,f)"

proof -

{ fix V assume "V∈?S"

then obtain U where "U∈T" and "V = f``(U)" by auto

hence "U ⊆ \<Union>T" and "f-``(V) = f-``(f``(U))" by auto

with `f ∈ inj(\<Union>T,Y)` `U∈T` have "f-``(V) ∈ T" using inj_vimage_image

by simp

} then show "IsContinuous(T,?S,f)" unfolding IsContinuous_def by auto

qed

ultimately show"IsAhomeomorphism(T,?S,f)" using bij_cont_open_homeo

by auto

qed

qed

section{*Partial functions and continuity*}

text{*Suppose we have two topologies $\tau_1,\tau_2$ on sets

$X_i=\bigcup\tau_i, i=1,2$. Consider some function $f:A\rightarrow X_2$, where

$A\subseteq X_1$ (we will call such function ''partial''). In such situation we have two

natural possibilities for the pairs of topologies with respect to which this function may

be continuous. One is obvously the original $\tau_1,\tau_2$ and in the second one

the first element of the pair is the topology relative to the domain of the

function: $\{A\cap U | U \in \tau_1\}$. These two possibilities are not exactly

the same and the goal of this section is to explore the differences.*}

text{*If a function is continuous, then its restriction is continous in relative

topology.*}

lemma (in two_top_spaces0) restr_cont:

assumes A1: "A ⊆ X⇩_{1}" and A2: "f {is continuous}"

shows "IsContinuous(τ⇩_{1}{restricted to} A, τ⇩_{2},restrict(f,A))"

proof -

let ?g = "restrict(f,A)"

{ fix U assume "U ∈ τ⇩_{2}"

with A2 have "f-``(U) ∈ τ⇩_{1}" using IsContinuous_def by simp;

moreover from A1 have "?g-``(U) = f-``(U) ∩ A"

using fmapAssum func1_2_L1 by simp;

ultimately have "?g-``(U) ∈ (τ⇩_{1}{restricted to} A)"

using RestrictedTo_def by auto;

} then show ?thesis using IsContinuous_def by simp;

qed

text{*If a function is continuous, then it is continuous when we restrict

the topology on the range to the image of the domain.*}

lemma (in two_top_spaces0) restr_image_cont:

assumes A1: "f {is continuous}"

shows "IsContinuous(τ⇩_{1}, τ⇩_{2}{restricted to} f``(X⇩_{1}),f)"

proof -

have "∀U ∈ τ⇩_{2}{restricted to} f``(X⇩_{1}). f-``(U) ∈ τ⇩_{1}"

proof;

fix U assume "U ∈ τ⇩_{2}{restricted to} f``(X⇩_{1})"

then obtain V where "V ∈ τ⇩_{2}" and "U = V ∩ f``(X⇩_{1})"

using RestrictedTo_def by auto;

with A1 show "f-``(U) ∈ τ⇩_{1}"

using fmapAssum inv_im_inter_im IsContinuous_def

by simp

qed

then show ?thesis using IsContinuous_def by simp;

qed;

text{*A combination of @{text "restr_cont"} and @{text "restr_image_cont"}.*}

lemma (in two_top_spaces0) restr_restr_image_cont:

assumes A1: "A ⊆ X⇩_{1}" and A2: "f {is continuous}" and

A3: "g = restrict(f,A)" and

A4: "τ⇩_{3}= τ⇩_{1}{restricted to} A"

shows "IsContinuous(τ⇩_{3}, τ⇩_{2}{restricted to} g``(A),g)"

proof -

from A1 A4 have "\<Union>τ⇩_{3}= A"

using union_restrict by auto

have "two_top_spaces0(τ⇩_{3}, τ⇩_{2}, g)"

proof -

from A4 have

"τ⇩_{3}{is a topology}" and "τ⇩_{2}{is a topology}"

using tau1_is_top tau2_is_top

topology0_def topology0.Top_1_L4 by auto

moreover from A1 A3 `\<Union>τ⇩_{3}= A` have "g: \<Union>τ⇩_{3}-> \<Union>τ⇩_{2}"

using fmapAssum restrict_type2 by simp;

ultimately show ?thesis using two_top_spaces0_def

by simp;

qed

moreover from assms have "IsContinuous(τ⇩_{3}, τ⇩_{2}, g)"

using restr_cont by simp;

ultimately have "IsContinuous(τ⇩_{3}, τ⇩_{2}{restricted to} g``(\<Union>τ⇩_{3}),g)"

by (rule two_top_spaces0.restr_image_cont);

moreover note `\<Union>τ⇩_{3}= A`

ultimately show ?thesis by simp;

qed

text{*We need a context similar to @{text "two_top_spaces0"} but without

the global function $f:X_1\rightarrow X_2$. *}

locale two_top_spaces1 =

fixes τ⇩_{1}

assumes tau1_is_top: "τ⇩_{1}{is a topology}"

fixes τ⇩_{2}

assumes tau2_is_top: "τ⇩_{2}{is a topology}"

fixes X⇩_{1}

defines X1_def [simp]: "X⇩_{1}≡ \<Union>τ⇩_{1}"

fixes X⇩_{2}

defines X2_def [simp]: "X⇩_{2}≡ \<Union>τ⇩_{2}"

text{*If a partial function $g:X_1\supseteq A\rightarrow X_2$ is continuous with

respect to $(\tau_1,\tau_2)$, then $A$ is open (in $\tau_1$) and

the function is continuous in the relative topology.*}

lemma (in two_top_spaces1) partial_fun_cont:

assumes A1: "g:A->X⇩_{2}" and A2: "IsContinuous(τ⇩_{1},τ⇩_{2},g)"

shows "A ∈ τ⇩_{1}" and "IsContinuous(τ⇩_{1}{restricted to} A, τ⇩_{2}, g)"

proof -

from A2 have "g-``(X⇩_{2}) ∈ τ⇩_{1}"

using tau2_is_top IsATopology_def IsContinuous_def by simp

with A1 show "A ∈ τ⇩_{1}" using func1_1_L4 by simp

{ fix V assume "V ∈ τ⇩_{2}"

with A2 have "g-``(V) ∈ τ⇩_{1}" using IsContinuous_def by simp

moreover

from A1 have "g-``(V) ⊆ A" using func1_1_L3 by simp

hence "g-``(V) = A ∩ g-``(V)" by auto

ultimately have "g-``(V) ∈ (τ⇩_{1}{restricted to} A)"

using RestrictedTo_def by auto

} then show "IsContinuous(τ⇩_{1}{restricted to} A, τ⇩_{2}, g)"

using IsContinuous_def by simp

qed

text{*For partial function defined on open sets continuity in the whole

and relative topologies are the same.*}

lemma (in two_top_spaces1) part_fun_on_open_cont:

assumes A1: "g:A->X⇩_{2}" and A2: "A ∈ τ⇩_{1}"

shows "IsContinuous(τ⇩_{1},τ⇩_{2},g) <->

IsContinuous(τ⇩_{1}{restricted to} A, τ⇩_{2}, g)"

proof

assume "IsContinuous(τ⇩_{1},τ⇩_{2},g)"

with A1 show "IsContinuous(τ⇩_{1}{restricted to} A, τ⇩_{2}, g)"

using partial_fun_cont by simp

next

assume I: "IsContinuous(τ⇩_{1}{restricted to} A, τ⇩_{2}, g)"

{ fix V assume "V ∈ τ⇩_{2}"

with I have "g-``(V) ∈ (τ⇩_{1}{restricted to} A)"

using IsContinuous_def by simp

then obtain W where "W ∈ τ⇩_{1}" and "g-``(V) = A∩W"

using RestrictedTo_def by auto

with A2 have "g-``(V) ∈ τ⇩_{1}" using tau1_is_top IsATopology_def

by simp

} then show "IsContinuous(τ⇩_{1},τ⇩_{2},g)" using IsContinuous_def

by simp

qed

section{*Product topology and continuity*}

text{*We start with three topological spaces $(\tau_1,X_1), (\tau_2,X_2)$ and

$(\tau_3,X_3)$ and a function $f:X_1\times X_2\rightarrow X_3$. We will study

the properties of $f$ with respect to the product topology $\tau_1\times \tau_2$

and $\tau_3$. This situation is similar as in locale @{text "two_top_spaces0"}

but the first topological space is assumed to be a product of two topological spaces.

*}

text{*First we define a locale with three topological spaces.*}

locale prod_top_spaces0 =

fixes τ⇩_{1}

assumes tau1_is_top: "τ⇩_{1}{is a topology}"

fixes τ⇩_{2}

assumes tau2_is_top: "τ⇩_{2}{is a topology}"

fixes τ⇩_{3}

assumes tau3_is_top: "τ⇩_{3}{is a topology}"

fixes X⇩_{1}

defines X1_def [simp]: "X⇩_{1}≡ \<Union>τ⇩_{1}"

fixes X⇩_{2}

defines X2_def [simp]: "X⇩_{2}≡ \<Union>τ⇩_{2}"

fixes X⇩_{3}

defines X3_def [simp]: "X⇩_{3}≡ \<Union>τ⇩_{3}"

fixes η

defines eta_def [simp]: "η ≡ ProductTopology(τ⇩_{1},τ⇩_{2})"

text{*Fixing the first variable in a two-variable continuous function results in a

continuous function.*}

lemma (in prod_top_spaces0) fix_1st_var_cont:

assumes "f: X⇩_{1}×X⇩_{2}->X⇩_{3}" and "IsContinuous(η,τ⇩_{3},f)"

and "x∈X⇩_{1}"

shows "IsContinuous(τ⇩_{2},τ⇩_{3},Fix1stVar(f,x))"

using assms fix_1st_var_vimage IsContinuous_def tau1_is_top tau2_is_top

prod_sec_open1 by simp

text{*Fixing the second variable in a two-variable continuous function results in a

continuous function.*}

lemma (in prod_top_spaces0) fix_2nd_var_cont:

assumes "f: X⇩_{1}×X⇩_{2}->X⇩_{3}" and "IsContinuous(η,τ⇩_{3},f)"

and "y∈X⇩_{2}"

shows "IsContinuous(τ⇩_{1},τ⇩_{3},Fix2ndVar(f,y))"

using assms fix_2nd_var_vimage IsContinuous_def tau1_is_top tau2_is_top

prod_sec_open2 by simp

text{*Having two constinuous mappings we can construct a third one on the cartesian product

of the domains.*}

lemma cart_prod_cont:

assumes A1: "τ⇩_{1}{is a topology}" "τ⇩_{2}{is a topology}" and

A2: "η⇩_{1}{is a topology}" "η⇩_{2}{is a topology}" and

A3a: "f⇩_{1}:\<Union>τ⇩_{1}->\<Union>η⇩_{1}" and A3b: "f⇩_{2}:\<Union>τ⇩_{2}->\<Union>η⇩_{2}" and

A4: "IsContinuous(τ⇩_{1},η⇩_{1},f⇩_{1})" "IsContinuous(τ⇩_{2},η⇩_{2},f⇩_{2})" and

A5: "g = {⟨p,⟨f⇩_{1}`(fst(p)),f⇩_{2}`(snd(p))⟩⟩. p ∈ \<Union>τ⇩_{1}×\<Union>τ⇩_{2}}"

shows "IsContinuous(ProductTopology(τ⇩_{1},τ⇩_{2}),ProductTopology(η⇩_{1},η⇩_{2}),g)"

proof -

let ?τ = "ProductTopology(τ⇩_{1},τ⇩_{2})"

let ?η = "ProductTopology(η⇩_{1},η⇩_{2})"

let ?X⇩_{1}= "\<Union>τ⇩_{1}"

let ?X⇩_{2}= "\<Union>τ⇩_{2}"

let ?Y⇩_{1}= "\<Union>η⇩_{1}"

let ?Y⇩_{2}= "\<Union>η⇩_{2}"

let ?B = "ProductCollection(η⇩_{1},η⇩_{2})"

from A1 A2 have "?τ {is a topology}" and "?η {is a topology}"

using Top_1_4_T1 by auto

moreover have "g: ?X⇩_{1}×?X⇩_{2}-> ?Y⇩_{1}×?Y⇩_{2}"

proof -

{ fix p assume "p ∈ ?X⇩_{1}×?X⇩_{2}"

hence "fst(p) ∈ ?X⇩_{1}" and "snd(p) ∈ ?X⇩_{2}" by auto

from A3a `fst(p) ∈ ?X⇩_{1}` have "f⇩_{1}`(fst(p)) ∈ ?Y⇩_{1}"

by (rule apply_funtype)

moreover from A3b `snd(p) ∈ ?X⇩_{2}` have "f⇩_{2}`(snd(p)) ∈ ?Y⇩_{2}"

by (rule apply_funtype)

ultimately have "⟨f⇩_{1}`(fst(p)),f⇩_{2}`(snd(p))⟩ ∈ \<Union>η⇩_{1}×\<Union>η⇩_{2}" by auto

} hence "∀p ∈ ?X⇩_{1}×?X⇩_{2}. ⟨f⇩_{1}`(fst(p)),f⇩_{2}`(snd(p))⟩ ∈ ?Y⇩_{1}×?Y⇩_{2}"

by simp

with A5 show "g: ?X⇩_{1}×?X⇩_{2}-> ?Y⇩_{1}×?Y⇩_{2}" using ZF_fun_from_total

by simp

qed

moreover from A1 A2 have "\<Union>?τ = ?X⇩_{1}×?X⇩_{2}" and "\<Union>?η = ?Y⇩_{1}×?Y⇩_{2}"

using Top_1_4_T1 by auto

ultimately have "two_top_spaces0(?τ,?η,g)" using two_top_spaces0_def

by simp

moreover from A2 have "?B {is a base for} ?η" using Top_1_4_T1

by simp

moreover have "∀U∈?B. g-``(U) ∈ ?τ"

proof

fix U assume "U∈?B"

then obtain V W where "V ∈ η⇩_{1}" "W ∈ η⇩_{2}" and "U = V×W"

using ProductCollection_def by auto

with A3a A3b A5 have "g-``(U) = f⇩_{1}-``(V) × f⇩_{2}-``(W)"

using cart_prod_fun_vimage by simp

moreover from A1 A4 `V ∈ η⇩_{1}` `W ∈ η⇩_{2}` have "f⇩_{1}-``(V) × f⇩_{2}-``(W) ∈ ?τ"

using IsContinuous_def prod_open_open_prod by simp

ultimately show "g-``(U) ∈ ?τ" by simp

qed

ultimately show ?thesis using two_top_spaces0.Top_ZF_2_1_L5

by simp

qed

text{*A reformulation of the @{text "cart_prod_cont"} lemma above in slightly different notation.*}

theorem (in two_top_spaces0) product_cont_functions:

assumes "f:X⇩_{1}->X⇩_{2}" "g:\<Union>τ⇩_{3}->\<Union>τ⇩_{4}"

"IsContinuous(τ⇩_{1},τ⇩_{2},f)" "IsContinuous(τ⇩_{3},τ⇩_{4},g)"

"τ⇩_{4}{is a topology}" "τ⇩_{3}{is a topology}"

shows "IsContinuous(ProductTopology(τ⇩_{1},τ⇩_{3}),ProductTopology(τ⇩_{2},τ⇩_{4}),{⟨⟨x,y⟩,⟨f`x,g`y⟩⟩. ⟨x,y⟩∈X⇩_{1}×\<Union>τ⇩_{3}})"

proof -

have "{⟨⟨x,y⟩,⟨f`x,g`y⟩⟩. ⟨x,y⟩∈X⇩_{1}×\<Union>τ⇩_{3}} = {⟨p,⟨f`(fst(p)),g`(snd(p))⟩⟩. p ∈ X⇩_{1}×\<Union>τ⇩_{3}}"

by force

with tau1_is_top tau2_is_top assms show ?thesis using cart_prod_cont by simp

qed

text{*A special case of @{text "cart_prod_cont"} when the function acting on the second

axis is the identity.*}

lemma cart_prod_cont1:

assumes A1: "τ⇩_{1}{is a topology}" and A1a: "τ⇩_{2}{is a topology}" and

A2: "η⇩_{1}{is a topology}" and

A3: "f⇩_{1}:\<Union>τ⇩_{1}->\<Union>η⇩_{1}" and A4: "IsContinuous(τ⇩_{1},η⇩_{1},f⇩_{1})" and

A5: "g = {⟨p, ⟨f⇩_{1}`(fst(p)),snd(p)⟩⟩. p ∈ \<Union>τ⇩_{1}×\<Union>τ⇩_{2}}"

shows "IsContinuous(ProductTopology(τ⇩_{1},τ⇩_{2}),ProductTopology(η⇩_{1},τ⇩_{2}),g)"

proof -

let ?f⇩_{2}= "id(\<Union>τ⇩_{2})"

have "∀x∈\<Union>τ⇩_{2}. ?f⇩_{2}`(x) = x" using id_conv by blast

hence I: "∀p ∈ \<Union>τ⇩_{1}×\<Union>τ⇩_{2}. snd(p) = ?f⇩_{2}`(snd(p))" by simp

note A1 A1a A2 A1a A3

moreover have "?f⇩_{2}:\<Union>τ⇩_{2}->\<Union>τ⇩_{2}" using id_type by simp

moreover note A4

moreover have "IsContinuous(τ⇩_{2},τ⇩_{2},?f⇩_{2})" using id_cont by simp

moreover have "g = {⟨p, ⟨f⇩_{1}`(fst(p)),?f⇩_{2}`(snd(p))⟩ ⟩. p ∈ \<Union>τ⇩_{1}×\<Union>τ⇩_{2}}"

proof

from A5 I show "g ⊆ {⟨p, ⟨f⇩_{1}`(fst(p)),?f⇩_{2}`(snd(p))⟩⟩. p ∈ \<Union>τ⇩_{1}×\<Union>τ⇩_{2}}"

by auto

from A5 I show "{⟨p, ⟨f⇩_{1}`(fst(p)),?f⇩_{2}`(snd(p))⟩⟩. p ∈ \<Union>τ⇩_{1}×\<Union>τ⇩_{2}} ⊆ g"

by auto

qed

ultimately show ?thesis by (rule cart_prod_cont)

qed

section{*Pasting lemma*}

text{*The classical pasting lemma states that if $U_1,U_2$ are both open (or closed) and a function

is continuous when restricted to both $U_1$ and $U_2$ then it is continuous

when restricted to $U_1 \cup U_2$. In this section we prove a generalization statement

stating that the set $\{ U \in \tau_1 | f|_U$ is continuous $\}$ is a

topology. *}

text{*A typical statement of the pasting lemma uses the notion of a function

restricted to a set being continuous without specifying the topologies with

respect to which this continuity holds.

In @{text "two_top_spaces0"} context the notation @{text "g {is continuous}"}

means continuity wth respect to topologies $\tau_1, \tau_2$.

The next lemma is a special case of @{text "partial_fun_cont"} and states that if

for some set $A\subseteq X_1=\bigcup \tau_1$

the function $f|_A$ is continuous (with respect to $(\tau_1, \tau_2)$), then

$A$ has to be open. This clears up terminology and indicates why we need

to pay attention to the issue of which topologies we talk about when we say

that the restricted (to some closed set for example) function is continuos.

*}

lemma (in two_top_spaces0) restriction_continuous1:

assumes A1: "A ⊆ X⇩_{1}" and A2: "restrict(f,A) {is continuous}"

shows "A ∈ τ⇩_{1}"

proof -

from assms have "two_top_spaces1(τ⇩_{1},τ⇩_{2})" and

"restrict(f,A):A->X⇩_{2}" and "restrict(f,A) {is continuous}"

using tau1_is_top tau2_is_top two_top_spaces1_def fmapAssum restrict_fun

by auto

then show ?thesis using two_top_spaces1.partial_fun_cont by simp

qed

text{*If a fuction is continuous on each set of a collection of open sets, then

it is continuous on the union of them. We could use continuity with respect to

the relative topology here, but we know that on open sets this is the same as the

original topology.*}

lemma (in two_top_spaces0) pasting_lemma1:

assumes A1: "M ⊆ τ⇩_{1}" and A2: "∀U∈M. restrict(f,U) {is continuous}"

shows "restrict(f,\<Union>M) {is continuous}"

proof -

{ fix V assume "V∈τ⇩_{2}"

from A1 have "\<Union>M ⊆ X⇩_{1}" by auto

then have "restrict(f,\<Union>M)-``(V) = f-``(V) ∩ (\<Union>M)"

using func1_2_L1 fmapAssum by simp

also have "… = \<Union> {f-``(V) ∩ U. U∈M}" by auto

finally have "restrict(f,\<Union>M)-``(V) = \<Union> {f-``(V) ∩ U. U∈M}" by simp

moreover

have "{f-``(V) ∩ U. U∈M} ∈ Pow(τ⇩_{1})"

proof -

{ fix W assume "W ∈ {f-``(V) ∩ U. U∈M}"

then obtain U where "U∈M" and I: "W = f-``(V) ∩ U" by auto

with A2 have "restrict(f,U) {is continuous}" by simp

with `V∈τ⇩_{2}` have "restrict(f,U)-``(V) ∈ τ⇩_{1}"

using IsContinuous_def by simp

moreover from `\<Union>M ⊆ X⇩_{1}` and `U∈M`

have "restrict(f,U)-``(V) = f-``(V) ∩ U"

using fmapAssum func1_2_L1 by blast

ultimately have "f-``(V) ∩ U ∈ τ⇩_{1}" by simp

with I have "W ∈ τ⇩_{1}" by simp

} then show ?thesis by auto

qed

then have "\<Union>{f-``(V) ∩ U. U∈M} ∈ τ⇩_{1}"

using tau1_is_top IsATopology_def by auto

ultimately have "restrict(f,\<Union>M)-``(V) ∈ τ⇩_{1}"

by simp

} then show ?thesis using IsContinuous_def by simp

qed

text{*If a function is continuous on two sets, then it is continuous

on intersection.*}

lemma (in two_top_spaces0) cont_inter_cont:

assumes A1: "A ⊆ X⇩_{1}" "B ⊆ X⇩_{1}" and

A2: "restrict(f,A) {is continuous}" "restrict(f,B) {is continuous}"

shows "restrict(f,A∩B) {is continuous}"

proof -

{ fix V assume "V∈τ⇩_{2}"

with assms have

"restrict(f,A)-``(V) = f-``(V) ∩ A" "restrict(f,B)-``(V) = f-``(V) ∩ B" and

"restrict(f,A)-``(V) ∈ τ⇩_{1}" and "restrict(f,B)-``(V) ∈ τ⇩_{1}"

using func1_2_L1 fmapAssum IsContinuous_def by auto

then have "(restrict(f,A)-``(V)) ∩ (restrict(f,B)-``(V)) = f-``(V) ∩ (A∩B)"

by auto

moreover

from A2 `V∈τ⇩_{2}` have

"restrict(f,A)-``(V) ∈ τ⇩_{1}" and "restrict(f,B)-``(V) ∈ τ⇩_{1}"

using IsContinuous_def by auto

then have "(restrict(f,A)-``(V)) ∩ (restrict(f,B)-``(V)) ∈ τ⇩_{1}"

using tau1_is_top IsATopology_def by simp

moreover

from A1 have "(A∩B) ⊆ X⇩_{1}" by auto

then have "restrict(f,A∩B)-``(V) = f-``(V) ∩ (A∩B)"

using func1_2_L1 fmapAssum by simp

ultimately have "restrict(f,A∩B)-``(V) ∈ τ⇩_{1}" by simp

} then show ?thesis using IsContinuous_def by auto

qed

text{*The collection of open sets $U$ such that $f$ restricted to

$U$ is continuous, is a topology.*}

theorem (in two_top_spaces0) pasting_theorem:

shows "{U ∈ τ⇩_{1}. restrict(f,U) {is continuous}} {is a topology}"

proof -

let ?T = "{U ∈ τ⇩_{1}. restrict(f,U) {is continuous}}"

have "∀M∈Pow(?T). \<Union>M ∈ ?T"

proof

fix M assume "M ∈ Pow(?T)"

then have "restrict(f,\<Union>M) {is continuous}"

using pasting_lemma1 by auto

with `M ∈ Pow(?T)` show "\<Union>M ∈ ?T"

using tau1_is_top IsATopology_def by auto

qed

moreover have "∀U∈?T.∀V∈?T. U∩V ∈ ?T"

using cont_inter_cont tau1_is_top IsATopology_def by auto

ultimately show ?thesis using IsATopology_def by simp

qed

text{*0 is continuous.*}

corollary (in two_top_spaces0) zero_continuous: shows "0 {is continuous}"

proof -

let ?T = "{U ∈ τ⇩_{1}. restrict(f,U) {is continuous}}"

have "?T {is a topology}" by (rule pasting_theorem)

then have "0∈?T" by (rule empty_open)

hence "restrict(f,0) {is continuous}" by simp

moreover have "restrict(f,0) = 0" by simp

ultimately show ?thesis by simp

qed

end