Topological Spaces∗

Topological Spaces∗ Alejandro Francetich† Department of Decision Sciences and IGIER Bocconi University, Italy

Topological spaces; open and closed sets. Bases. Subspaces. Interior and closure of a set. Compact and connected spaces. Continuous functions.

∗ These

notes draw heavily from (in alphabetical order) Munkres (2000) and Rodriguez (2007). The notes are work in progress, and subject to changes. Comments, suggestions, and typo notifications are encouraged. † Via Röntgen 1, 20136 Milan, Italy. Email address: [email protected]

Contents 1

Introduction

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2

Topological Spaces: Open and Closed Sets

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3

Interior and Closure of Arbitrary Sets

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4

Compact and Connected Spaces

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5

Continuity of Functions

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Introduction In Euclidean spaces, we defined open sets, and showed the following: • the entire space and the empty set are open; • arbitrary unions of open sets are open; • finite intersections of open sets are open. Having a notion of openness allows us to talk about continuity of functions, among

other things. Now, the notion of openness in Euclidean spaces is actually more general than it may seem. For instance, if we go back to the definition of open sets for Euclidean spaces and replace the Euclidean distance with the “taxi-cab metric,” given by d1 ( x, y) := ∑kj=1 | x j − y j |, the same collection of subsets obtains: A subset of R k is open if and only if it is open under this alternative metric. Moreover, in many problems in Economics, we want to be able to talk about these notions in more general spaces than Euclidean spaces. For instance, in the context of choice under uncertainty, we want to study utility functions defined on sets of probability distributions. What does it mean for such a utility function to be “continuous”? The approach we shall follow is, given some arbitrary set, we specify (as primitive) which of its subsets are to be considered “open.” This leads to the notion of a topological space: a set paired with a collection of subsets (the “open” subsets). The three properties that we derived for open sets in Euclidean spaces are imposed as axioms.

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Topological Spaces: Open and Closed Sets Let X be an arbitrary non-empty set, and let P ( X ) := { A : A ⊆ X } denote the power

set of X, namely, the set of all subsets X. A topology on X, denoted by T ⊆ P ( X ), is a collection of subsets of X that satisfies the following axioms. 1

1. Both X and ∅ belong to T . 2. If Δ is a set of indices, and {Oδ ∈ T : δ ∈ Δ} is an indexed family of sets in T , then

∪δ∈Δ Oδ ∈ T . (Closure under arbitrary union) 3. If Δ is a finite set of indices, and {Oδ ∈ T : δ ∈ Δ} is an indexed family of sets in

T , then ∩δ∈Δ Oδ ∈ T . (Closure under finite intersection) The pair ( X, T ) is called a topological space. In this topological space, a set O ⊆ X is open if O ∈ T ; while a set C ⊆ X is closed if C c ∈ T .

The properties of open sets presented in Proposition 1 in Unit 1 are now introduced as axioms. The corresponding properties of closed sets in Proposition 2 follow by the exact same argument. Example 1. Consider the set X = { a, b, c}. The following collection of sets is a topology

on X: T = {∅, { a}, {b, c}, X }.

Now, the three axioms that define a topology do not uniquely determine a topology. Indeed, for any arbitrary set, we have at least two topologies: the trivial topology, T0 :=

{∅, X }, and the discrete topology, T1 := P ( X ). For Euclidean spaces, the collection of

sets defined as open in Unit 1 is called the usual topology; it is the default topology when dealing with vectors of real numbers. Given two topologies T1 , T2 on a set X, we say that T1 is finer or stronger than T2 , and that T2 is coarser or weaker than T1 , if T2 ⊆ T1 . The trivial topology is the coarsest or weakest topology for any space, while the discrete topology is the finest or strongest. Any other topology lies in between these two. Given a topological space ( X, T ) and a point x ∈ X, a subset N ⊆ X is a neighborhood of x if there exists an open set U ∈ T such that x ∈ U ⊆ N. In words, a neighborhood of a point is a set that contains an open to which this point belongs. If the set N is itself open, then it is an open neighborhood of x. 2

The notion of neighborhoods leads to the following characterization of open sets, which is reminiscent of the definition of open sets for Euclidean spaces: Open sets are precisely those who contain neighborhoods of all of their points. Proposition 1. Let ( X, T ) be a topological space. A set A ⊆ X is open if and only if, for every x ∈ A, there exists a neighborhood Nx of x such that Nx ⊆ A. Proof. If A ∈ T , then A itself is an (open) neighborhood of every x ∈ A. Conversely, for each x ∈ A, let Nx be a neighborhood of x contained in A, and let Ox be an open set such that x ∈ Ox ⊆ Nx ⊆ A. We can write A as the union of all of these open sets: A = ∪ x∈ A { x } ⊆ ∪ x∈ A Ox ⊆ ∪ x∈ A Nx ⊆ A, so A = ∪ x∈ A Ox . As arbitrary unions of open sets are open, it follows that A ∈ T . Bases for topologies We can specify a topology on a set by listing all of the subsets that are to be considered open, or by specifying a definition for open set (as we did for Euclidean spaces). Yet another way is to specify a collection of sets from which the open sets are to be obtained, by performing certain operations. Let ( X, T ) be a topological space. A collection BT ⊆ T of open subsets of X is a basis for T if, for every O ∈ T , there exists some index set ΔO and some sub-collection of sets

{ Bδ : δ ∈ ΔO } ⊆ BT such that O = ∪δ∈ΔO Bδ . The open sets in BT are called basic sets.

All of the open subsets of X can be obtained as unions of basic open sets. Example 2. The collection of open intervals B = {( a, b) : a, b ∈ R, b > a} is a basis for the usual topology on the real line. Recall that a set A ⊆ R is open if, for each x ∈ A, we

can find some ex > 0 such that ( x − ex , x + ex ) ⊆ A. By a similar chain of set inclusions

as in the proof of Proposition 1, we find that A = ∪ x∈ A ( x − ex , x + ex ). 3

A topology can be identified by means of different bases. However, a basis corresponds to a unique topology.

Proposition 2. Let T , T 0 be two topologies on a set X, and let B be a basis for both of these

topologies. Then, T = T 0 .

Proof. Let O ∈ T . Since B is a basis for T , we can find an indexed family of sets in B ,  O Bδ ∈ B : δ ∈ Δ , such that O = ∪δ∈Δ BδO . Since B is also a basis for T 0 , we have that

B ⊆ T 0 . As a union of sets that are open under T 0 , we have that O ∈ T 0 . The converse

inclusion follows immediately by reversing the roles of T and T 0 . If we start with some arbitrary collection B ⊆ P ( X ), we can always construct a topology on X that contains B .

The construction goes as follows. Let F (B) be the collection of topologies that contain

B . Since P ( X ) ∈ F (B), this collection is non-empty. Moreover, this collection is closed

under arbitrary intersections. To see this, let Δ be a non-empty index set, and consider a collection {Tδ : δ ∈ Δ} ⊆ F (B). Since B ⊆ Tδ for each δ ∈ Δ, it follows that B ⊆ ∩δ∈Δ Tδ . As ∅, X ∈ Tδ for all δ ∈ Δ, we have ∅, X ∈ ∩δ∈Δ Tδ . Now, let O be a collection of sets in

∩δ∈Δ Tδ . For each δ ∈ Δ, O is contained in Tδ , so ∪O∈O O ∈ Tδ . Thus, ∪O∈O O ∈ ∩δ∈Δ Tδ .

Similarly, if O 0 is a finite collection of sets in ∩δ∈Δ Tδ , then ∩O∈O 0 O ∈ Tδ for all δ ∈ Δ; hence, ∩O∈O 0 O ∈ ∩δ∈Δ Tδ . We conclude that ∩δ∈Δ Tδ is a topology. The topology given by

TB := ∩T ∈F (B) T is called the topology generated by B . By construction, it is the coarsest

topology that contains B . In general, B need not be a basis for TB ; we may find some open sets that cannot be written as unions of elements of B . Example 3. Consider the collection of open intervals on R given by: B = { I (n) :=

(n − 1, n + 1) : n ∈ N }, and consider the set A = (3, 4). This set cannot be written as 4

the union of elements in B ; the only candidates are I (3) = (2, 4) and I (4) = (3, 5), but they each contain points not in A. However, A is open under any topology that contains

B , since A = I (3) ∩ I (4) is a finite intersection of elements in B . Therefore, A ∈ TB , but A is not a union of elements of B . The problem in this example is that B is not “rich enough” to be able to express all of the sets in TB as unions of sets in B . However, if the collection B is a basis for some topology T , then it is also a basis for TB . To see this, notice that any O ∈ TB is also in  T . Therefore, we can find an indexed family of sets in B , BδO ∈ B : δ ∈ Δ , such that

O = ∪δ∈Δ BδO . Proposition 2 establishes that, in this case, TB = T . In words, when the collection B is rich enough to be the basis of a (unique) topology, then the topology that has B as a basis is the topology generated by B . Topological subspaces We may be interested in a subset of the space rather than on the entire space. When the entire space has a topological structure, there is a natural way in which we can view its subsets as topological spaces. This leads to the notions of relative openness as closedness, presented for Euclidean spaces in Unit 1.

Let S ⊆ X be a subset of a topological space ( X, T ). Denote by TS the collection of subsets of S that is obtained by intersecting all of the open subsets of X with S:

TS := {O ∩ S : O ∈ T }. The next proposition establishes that TS is a topology on S. Proposition 3. Let ( X, T ) be a topological space, and let S ⊆ X be any subset of X. The collection TS is a topology on S. Proof. Since ∅ ∈ T and ∅ = ∅ ∩ S, we have ∅ ∈ TS . Similarly, X ∈ T and S = X ∩ S,  so S ∈ TS . Let OδS ∈ TS : δ ∈ Δ be an indexed family of sets in TS , where Δ is a set

of indices. Then, there is a collection {Oδ ∈ T : δ ∈ Δ} such that OδS = Oδ ∩ S for each 5

δ ∈ Δ. By properties of T , we have ∪δ∈Δ Oδ ∈ T , so ∪δ∈Δ OδS = (∪δ∈Δ Oδ ) ∩ S ∈ TS .  Finally, let OδS ∈ TS : δ ∈ Δ be an indexed family of sets in TS , where Δ is now a finite

set of indices. We have a finite collection {Oδ ∈ T : δ ∈ Δ} such that OδS = Oδ ∩ S for all δ ∈ Δ. Then, ∩δ∈Δ Oδ ∈ T and ∩δ∈Δ OδS = (∩δ∈Δ Oδ ) ∩ S ∈ TS .

The topology TS is called the relative topology on S, and the space (S, TS ) is a topological

subspace of ( X, T ). A subset A ⊆ S is relatively open (in S) if A ∈ TS , and it is relatively closed (in S) if the relative complement of A in S is relatively open: B \ A ∈ TS . The following is a simple characterization of relatively-closed sets. Proposition 4. Let ( B, T B ) be a topological subspace of ( X, T ). A set A ⊆ B is relatively closed if and only if we can find a closed subset of X, call it C, such that A = C ∩ B.

Proof. By definition, A is relatively closed if B \ A = B ∩ Ac is relatively open. Thus, we

can find some set O ∈ T such that B ∩ Ac = O ∩ B. By De Morgan’s Laws, Bc ∪ A =

Oc ∪ Bc . Taking intersection with B gives B ∩ ( Bc ∪ A) = ( B ∩ Bc ) ∪ ( B ∩ A) = A, and B ∩ (Oc ∪ Bc ) = ( B ∩ Oc ) ∪ ( B ∩ Bc ) = Oc ∩ B. So, A = Oc ∩ B, where Oc is a closed

subset of X. Conversely, let C be a closed subset of X such that A = C ∩ B. Again by De Morgan’s Laws, B \ A = (C c ∪ Bc ) ∩ B = C c ∩ B. This implies that B \ A ∈ T B .

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Interior and Closure of Arbitrary Sets In many topological spaces, some sets are neither open nor closed. This is the case

of intervals of the form [ a, b), for real numbers a < b, in the real line. However, in any topological space, we can correspond an open and a closed set to any set. Consider a topological space ( X, T ), and take an arbitrary subset S ⊆ X. Consider

the family of open sets contained in S: O(S) := {O ⊆ S : O ∈ T }. Since ∅ ⊆ S and ∅ ∈ T , the collection O(S) is non empty. Moreover, by the second axiom of topologies, 6

any arbitrary union of sets in O(S) is in itself a set in O(S). The interior of S, int(S),

is the union of all sets in O(S): int(S) := ∪O∈O(S) O. In words, the interior of S is the largest open set contained in S (largest in the sense of set inclusion). Example 4. Let a, b ∈ R, a < b. Under the usual topology, the interior of [ a, b) is ( a, b). Some properties of the interior of a set are the following. 1. For any set A ⊆ X, A ∈ T if and only if A = int( A). 2. For any A, B ⊆ X such that A ⊆ B, int( A) ⊆ int( B). 3. For any A, B ⊆ X, int( A) ∪ int( B) ⊆ int( A ∪ B). 4. For any A, B ⊆ X, int( A ∩ B) = int( A) ∩ int( B).

Similarly, consider the family of closed sets that contain S: C(S) := {S ⊆ C : C c ∈ T }.

Since S ⊆ X and X c = ∅ ∈ T , C(S) is non empty. Moreover, any arbitrary intersection

of sets in C(S) is in C(S). The closure of S, cl(S), is the intersection of all sets in C(S): cl(S) := ∩C∈C(S) C. It is the smallest closed set that contains S.1

Example 5. Let a, b ∈ R, a < b. Under the usual topology, the closure of [ a, b) is [ a, b]. Some properties of the closure of a set are the following. 1. For any set A ⊆ X, Ac ∈ T if and only if A = cl( A). 2. For any A, B ⊆ X such that A ⊆ B, cl( A) ⊆ cl( B). 3. For any A, B ⊆ X, cl( A ∪ B) = cl( A) ∪ cl( B). 4. For any A, B ⊆ X, cl( A ∩ B) ⊆ cl( A) ∩ cl( B). 1 The

connection between this notion of closure and the notion presented in Unit 1 will be made clear in Unit 3, when we study metric spaces.

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Of course, it may happen that the interior of a set is empty; for instance, in the real line, int({0}) = ∅. On the other extreme, there are sets whose closure is the entire space

X. These sets are called dense. Formally, a set A ⊆ X is dense in X if cl( A) = X. An

important example of a dense set is the set of rationals, Q, as a subset of the real line. When the entire space has a dense subset which is countable, as in this example, we say that the space is separable.

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Compact and Connected Spaces In Euclidean spaces, we defined compact sets as sets that are both closed and bounded.

This definition is not useful for arbitrary topological spaces, as we may not have a notion of boundedness. To define compactness in arbitrary topological spaces, we need some additional definitions. In a topological space ( X, T ), an open cover of X is a collection O ⊆ T of open sets

whose union equals X: X = ∪O∈O O. If, in addition, O is finite, then it is called a finite cover of X. If O and U are two open covers of X such that U ⊆ O , then U is an open

subcover of O . We say that the space ( X, T ) is compact if every open cover of X has a finite subcover. A subset A ⊆ X is compact of X is the subspace ( A, T A ) is compact.

Compactness can be thought of as a generalization of finiteness, in the following sense: Finite sets in any topological space are compact, real-valued functions have maxima and minima on compact sets. The latter result is a generalization of Weierstrass’ Theorem that follows from Proposition 9 bellow. The former result is given in the next proposition. Proposition 5. Let ( X, T ) be a topological space. If A ⊆ X is finite, then A is compact. Proof. Let O A be a relatively open cover of A. For each x ∈ A, we can find some O xA ∈ O A  such that x ∈ O xA . The collection O xA : x ∈ A is a finite subcover of O A . 8

This definition may look very different from the definition of compact sets given for Euclidean spaces. In general, it is: There are compact topological spaces are not bounded, as well as compact spaces that are not closed.

Example 6. Consider an infinite set A ⊆ R endowed with the co-finite topology, given by

Tc f := {O ⊆ A : O = ∅ or A \ O is finite}. Under this topology, any non-empty open

set excludes, at most, finitely many points in A. Let B be an infinite subset of A. Take an open cover O of B, and pick any non-empty O ∈ O ; keep this choice fixed. We have that

B \ O ⊆ A \ O is finite, so we can write B \ O = { x1 , . . . , xn } for some n ∈ N. For each

i = 1, . . . , n, xi ∈ B, so we can find some Oi ∈ O such that xi ∈ Oi . It follows that the finite collection {O, O1 , . . . , On } is a finite subcover of O . We conclude that any infinite subset of A is compact under this topology. In particular, all the open subsets of A are

compact. However, any non-empty proper subset of A that is closed is finite: If A \ B is

non-empty and open, then B = A \ ( A \ B) must be finite. Yet the only finite set that is open is the empty set.

Nonetheless, examples of the latter are “weird”; for Euclidean spaces, the two notions coincide. This result is known as the Heine-Borel Theorem; its proof is beyond the scope of the course, so we shall only state this deep result. The characterization of compact sets as closed and bounded extends to somewhat more general spaces as well. We shall delve deeper into this in the coming units. Proposition 6 (The Heine-Borel Theorem). Let k ∈ N, and let A ⊆ R k . Set A is compact if and only if it is closed and bounded.

The next proposition presents a link between closedness and compactness that applies to any topological space.

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Proposition 7. Let ( X, T ) be a compact topological space. If a subset Y ⊆ X is closed, then it is compact. Proof. Let OY be an open cover of Y under the relative topology, TY . By definition of

the relative topology, we can write this cover as OY := {Oδ ∩ Y : δ ∈ Δ}, where Δ is an

index set and where Oδ ∈ T for all δ ∈ Δ. Since Y is closed, we have that Y c ∈ T . Now,

the collection O := {Oδ : δ ∈ Δ} ∪ {Y c } is an open cover of X. By compactness, there

exists a finite subset of Δ, Δ0 ⊆ Δ, such that {Oδ : δ ∈ Δ0 } ∪ {Y c } is a subcover. Then,

{Oδ ∩ Y : δ ∈ Δ0 } is a finite subcover of OY .

As for connectedness, the same notion introduced for Euclidean spaces applies to arbitrary topological spaces. A topological space ( X, T ) is a disconnected space if we can find two nonempty open sets A, B ∈ T such that X = A ∪ B. A space that is not

disconnected is called a connected space. A subset A ⊆ X is a connected subset of X if the

subspace ( A, T A ) is a connected space.

Now, the definitions of compact and connected subsets refer to the relative topology. The next proposition shows that need not worry about relativizing the topology. Proposition 8. Let ( X, T ) be a topological space. For any subset A ⊆ X, 1. A is a compact subset of X if and only if, for any collection O ⊆ T of open subsets of X such that A ⊆ ∪O∈O O, there exists a finite subcollection O 0 ⊆ O such that A ⊆ ∪O∈O 0 O;

2. A is a disconnected subset of X if and only if we can find two non-empty sets U , V ∈ T such that U ∩ V ∩ A = ∅, U ∩ A 6= ∅, V ∩ A 6= ∅, and (U ∩ A) ∪ (V ∩ A) = A.

Proof. (Compactness) Let A be a compact subset of X, and let O ⊆ T be such that A ⊆ ∪O∈O O. Then, the collection {O ∩ A : O ∈ O} is an open cover of A under the

relative topology. By compactness, there exists a finite subcollection O 0 ⊆ O such that

{O ∩ A : O ∈ O 0 } is a subcover. This implies that A ⊆ ∪O∈O 0 O, as desired. Conversely, 10

let O A be an open cover of A; we can write it as O A := {Oδ ∩ A : δ ∈ Δ}, for some index set Δ and a collection Oδ ∈ T , δ ∈ Δ of open subsets of X. We have that A ⊆ ∪δ∈Δ Oδ , so we can find a finite subset Δ0 ⊆ Δ such that A ⊆ ∪δ∈Δ0 Oδ . But then, {Oδ ∩ A : δ ∈ Δ0 }

is a finite subcover of O A . (Connectedness) Let A be a disconnected subset of X; we can produce two non-empty, relatively open sets U A , VA ∈ T A such that A = U A ∪ VA . By definition of the relative

topology, we can produce two non-empty sets U, V ∈ T such that U A = U ∩ A and

VA = V ∩ A. Now, U ∩ V ∩ A = U A ∩ VA = ∅; U ∩ A = U A 6= ∅, V ∩ A = VA 6= ∅; and

(U ∩ A) ∪ (V ∩ A) = U A ∪ VA = A, as desired. Conversely, assume that we can produce

two open subsets U, V of X with the properties in the statement of the proposition. Then, the sets U A := U ∩ A and VA := V ∩ A are non-empty, disjoint, relatively open, and cover A: U A ∪ VA = A. Thus, A is disconnected.

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Continuity of Functions Consider two topological spaces, ( X, T X ) and (Y, TY ), a set D ⊆ X, and a function

f : D → Y. Let T X | D denote the relative topology on D. We say that f : D → Y

is continuous at x0 ∈ D if, for every open neighborhood Vf ( x0 ) ∈ TY of f ( x0 ), we can

find a relatively-open neighborhood Ux0 ∈ T X | D of x0 such that f (Ux0 ) ⊆ Vf ( x0 ) . If f is continuous at every point in some B ⊆ D, we say that f is continuous on B. Finally, if f is continuous on all of its domain, we simply say that f is continuous. A point x0 ∈ D is a discontinuity point of f if we can find an open neighborhood

Vf0( x

∈ TY of f ( x0 ) such that, for every relatively-open neighborhood Ux0 ∈ T X | D , we  c have that f (Ux0 ) ∩ Vf0( x ) 6= ∅. 0)

0

These definitions are reminiscent of the definitions given for real-valued functions on

Euclidean spaces; the only difference is that the open balls have been replaced by the more general notion of open neighborhood. But the idea of continuity remains the same:

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By looking at neighboring points in of x0 , we obtain neighboring points of f ( x0 ). As in Unit 1, we have several alternative ways to characterize continuous functions. Theorem 1. Let ( X, T X ) and (Y, TY ) be two topological spaces, and let f : D → Y be a function defined on some subset D ⊆ X. The following statements are equivalent. 1. The function f is continuous. 2. For any open subset O of Y, the set f −1 (O) is relatively open in D. 3. For any closed subset C of Y, the set f −1 (C ) is relatively closed in D. Proof. (1 ⇒ 2) Take any O ∈ TY , and take any x ∈ f −1 (O) ⊆ D. The set O is an

open neighborhood of f ( x ). By continuity, there exists a relatively-open neighborhood

Ux ∈ T X | D of x such that f (Ux ) ⊆ O, or Ux ⊆ f −1 (O). Proposition 1 implies that f −1 (O) is relatively open.

(2 ⇒ 1) Take any x ∈ D, and let Vf ( x) ∈ TY be an open neighborhood of f ( x ). We   have that f −1 Vf ( x) is relatively open in D; thus, it is a relatively-open neighborhood    of x. By properties of images of functions, it follows that f f −1 Vf ( x) ⊆ Vf ( x ) . (2 ⇒ 3) Let C be a closed subset of Y; then, C c ∈ TY . By continuity, f −1 (C c ) is

relatively open in D. Thus, f −1 (C ) = f −1 (C c )c is relatively closed in D.

(3 ⇒ 2) Let O be an open subset of Y; then, Oc is closed, so f −1 (Oc ) is relatively

closed in D. Thus, f −1 (O) = f −1 (Oc )c is open relative to D.

We now state and prove the following two important properties of continuous functions between topological spaces. Proposition 9. Let ( X, T X ) and (Y, TY ) be two topological spaces, and let f : D → Y be a function defined on some subset D ⊆ X. Consider a subset A ⊆ D. 1. If A is a compact subset of D, then f ( A) is a compact subset of Y. 12

2. If A is a connected subset of D, then f ( A) is a connected subset of Y. Proof. (1.) Let O ⊆ TY be a collection of open sets such that f ( A) ⊆ ∪O∈O O. By continuity of f , we have that { f −1 (O) : O ∈ O} is a collection of relatively-open sets

in D. Moreover, A ⊆ ∪O∈O f −1 (O); by compactness of A, there exists a finite sub-

collection O 0 ⊆ O such that A ⊆ ∪O∈O 0 f −1 (O). This subcollection is a subcover:

f ( A) ⊆ f ∪O∈O 0 f −1 (O) = ∪O∈O 0 f f −1 (O) ⊆ ∪O∈O 0 O. Thus, f ( A) is a compact

subset of Y.

(2.) Assume that f ( A) is disconnected; we can find two non-empty open subsets of Y, U, V ∈ TY , such that U ∩ V ∩ f ( A) = ∅, U ∩ f ( A) 6= ∅, V ∩ f ( A) 6= ∅, and

(U ∩ f ( A)) ∪ (V ∩ f ( A)) = f ( A). Since f is continuous, the sets f −1 (U ), f −1 (V ) are

relatively open in D. Moreover, we have f −1 (U ) ∩ f −1 (V ) ∩ A = ∅, f −1 (U ) ∩ A 6= ∅, f 1 (V ) ∩ A 6= ∅, and ( f −1 (U ) ∩ A) ∪ ( f −1 (V ) ∩ A) = A. Thus, A is disconnected.

If the co-domain is the real line, part (1) of Proposition 9 provides an immediate extension of Weierstrass’ Theorem to real-valued functions defined on compact domains in arbitrary topological spaces. Similarly, part (2) allows us to extend the IntermediateValue Theorem. Next follows a simple, yet important result on composition of continuous functions between topological spaces. Proposition 10. Let ( X, T X ), (Y, TY ), and ( Z, T Z ) be three topological spaces. Consider a set

D ⊆ X, a function f : D → Y, a set f ( D ) ⊆ V ⊆ Y, and a function g : V → Z. Let g ◦ f : D → Z be the composite function ( g ◦ f )( x ) := g( f ( x )). If f and g are continuous, then

g ◦ f is continuous. Proof. Let O be an open subset of Z. Since g is continuous, the set g−1 (O) is relatively open in V. Let U be an open subset of Y such that g−1 (O) = U ∩ V. Now, ( g ◦ f )−1 (O) =

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f −1 g−1 (O) = f −1 (U ∩ V ) = f −1 (U ) ∩ f −1 (V ) = f −1 (U ) ∩ D = f −1 (U ). Since f is

continuous, we conclude that ( g ◦ f )−1 (O) is relatively open in D.

Consider a function f between a set X and a topological space (Y, TY ). Given a

topology on X, f might not be continuous. Of course, if the topology is all of P ( X ), then f will be continuous. But we can always construct a “less permissive,” more interesting topology on X that makes f continuous. The topology on X generated by the collection of sets { f −1 (O) : O ∈ TY } makes f continuous. Moreover, it is the coarsest topology to do so. This topology is called the topology generated by f .

References Munkres, J. (2000). Topology. Prentice Hall. Rodriguez, E. (2007). Elementos de topologia para economia y gestion. Lecture notes, FCE UBA.

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