Lecture : Topological Space

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1 Example of Lecture : Dr. Department of Mathematics Lovely Professional University Punjab, India October 18, 2014

2 Outline Example of Example of 4 5 6

3 Example of I Topological spaces and continuous functions between them are the primary objects of study in the field of topology. In next 1 to 15 lectures, we introduce topological spaces and some important concepts associated with them, including open sets, bases, and closed sets, clouter and interior. We will present two applications of topological spaces 1 Involving digital image processing 2 Concerning evolutionary proximity in biology In this lecture we will introduce the dedition of and their examples.

4 Example of II For many years, prior to the formalization of the field of topology, mathematicians used the concept of an open set, a simple example of which is an open interval on the real line. But over time it was realized that many of the properties held by open sets on the real line could be said to hold for certain types of subsets in any set. Eventually, the essential properties were distilled out and the concept of a collection of open sets, called a topology, evolved into the definition of topological space.

5 Example of I A topological space is supposed to be a set that has just enough structure to meaningfully speak of continuous functions on it.

6 Example of II Definition Let X be a any non-empty set. A topology on X is a collection τ of subset of X such that: 1 The set X and empty set φ belong to τ. 2 The union of any arbitrary (finite or infinite) number of subsets of τ belongs to τ. In other words if G i belong to τ is arbitrary, then i I G i τ, where I is any set. 3 The intersection of finite many subsets of τ belong to τ. In other words assume that for each i I if G i belong to τ, then i I G i τ, where I is finite set.

7 Example of III In short, a topology on a set X is a collection of subsets of X which includes empty set and X and is closed under unions and finite intersections. The subsets of collection τ are called open. A set together with a topology is called a topological space and denoted by (X, τ). Question: Which components involve to make topological space? There are two things that make up topological space : a set X, and a collection, τ, of subsets of X that forms a topology on X. To be properly formal, we should refer to a topological space as an ordered pair (X, τ), but to simplify notation we follow the common practice of refereing to

8 Example of IV the set X as a topological space, leaving it implicitly understood that there is a topology on X.

9 Example of V Remark 1 The statement intersection of finitely many subsets of τ is equivalent to the statement intersection of two subsets of τ. 2 One should be careful when using the word open. Open intervals in R are generally not the same thing as open sets in a topological space. Open sets are a generalization of the concept of open intervals.

10 Example of Example of I Example If X be the three-point set X = {a, b, c}, the there are many possibilities of topologies on X. Here we considered some of them schematically in figure 1 The set X with topology τ1 = {X, φ}. 2 The set X with topology τ2 = {X, φ, {a}, {a, b}}. 3 The set X with topology τ3 = {X, φ, {b}, {a, b}, {a, c}}. 4 The set X with topology τ4 = {X, φ, {b}}. 5 The set X with topology τ5 = {X, φ, {a}, {b, c}}. 6 The set X with topology τ6 = {X, φ, {b}, {c}, {a, b}, {b, c}}. 7 The set X with topology τ7 = {X, φ, {a, c}}.

11 Example of Example of II 8 The set X with topology τ8 = {X, φ, {a}, {b}, {a, b}}. 9 The set X with topology τ 9 = {X, φ, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}}. On the set of X there are many topologies can be defined but all the collections of subsets of X are not necessary to be topologies on X. The collections {X, φ, {a}, {b}} and {X, φ, {a, b}, {b, c}} of subsets of X are not topologies on X.

12 Example of Example of III Example Let N be the set of all natural numbers (that is, the set of all positive integers) and let τ consist of N, φ and all finite subsets of N. Then collection τ is not a topology on N, because the infinite union {2} {3}... {n}... = {2, 3,..., n,...} of members of τ does not belong to τ.

13 Example of Example of IV We have now defined three different topologies on X; The non-discrete or trivial topology The discrete topology The finite complement topology In the case of these three, the trivial topology, with fewest open sets, is contained within the finite complement topology, which is itself contained within the discrete topology.

14 Example of Example of V Example (Non-discrete Topology) Let X be a non-empty set. Define τ = {φ, X}. Notice that τ satisfies all three of the conditions for being a topology. However, if we remove either set, we no longer have a topology. Thus {φ, X} is minimal topology we can define on X. For obvious reasons, it is called the trivial topology or non-discrete on X.

15 Example of Example of VI Example (Discrete Topology) Let X be a nonempty set and let τ be the collection of all subsets of X. Clearly this is a topology, since unions and intersections of subsets of X are themselves subsets of X and therefore are in the collection τ. We call this the discrete topology on X. This is the largest topology that we can define on X.

16 Example of Example of VII Example (Finite Complement Topology) Let X be any set and τ f is the collections of all subsets U of X such that X U either is finite or empty or all of X i.e. τ f = {U X : X U either finite or empty or all of X} then the collection τ f is topology on X and called finite complement topology on X. Here we can check that above collection is in fact a topology on X. 1 Since X X = φ is finite and X φ = X is all of X, so X and φ belong to τ f.

17 Example of Example of VIII 2 Let {Uα } be a collections of open sets of X i.e. member of τ f,then we want to show that U α is in τ f. And for this we will show that X U α is finite. X U α =X (U 1 U2...) =(X U 1 ) (X U 2 )... Since {U 1, U 2,...} are open so by the definition, {(X U 1 ), (X U 2 ),...} are finite and by the properties of finite set that intersection of finite sets are finite so {(X U 1 ) (X U 2 )...} is finite i.e. X U α is finite. Hence by the hypothesis, U α is in τ f. 3 Similarly, we can check for finite intersection of open sets.

18 Example of Example of IX Example (Complement topology) On the real line R, define a topology whose open sets are the empty set and every set in R with a finite complement. For example, U = R {0, 3, 7} is an open set. We call this topology complement topology on R and denote it by R fc.

19 Example of Example of X Example Consider R, the set of real numbers, with τ = {S R: ɛ > 0 s.t. (x ɛ, x + ɛ) S} Here τ is a topology on R called usual topology and (R, τ) called usual topological space. 1 By the definition of τ, clearly τ P (R). And so we can say that φ τ is vacuously true and also R τ.

20 Example of Example of XI 2 Second, let {Aα }, α I be a family of sets indexed over set I such that for all α I, A α τ. Now we want to show that α I A α τ. Let W = α I A α. For all x in W, there exists α in I such that x in A α. So, by hypothesis, there exists ɛ > 0 such that (x ɛ, x + ɛ) A α W. Therefore, α I A α τ. 3 Finally, let A, B in τ. We must show that A B in τ. By hypothesis, if A and B in τ, then there exists non-zero ɛ a and ɛ b such that (x ɛ a, x + ɛ a ) A and (x ɛ b, x + ɛ b ) B. Now choose ɛ = min{ɛ a, ɛ b }. Then (x ɛ, x + ɛ) A and (x ɛ, x + ɛ) B. So, we can say that (x ɛ, x + ɛ) A B. Therefore, A B in τ.

21 Example of Example of XII Example (Topology induced by the metric) Let (X, d) be a metric space. Define O X to be open if for any x in O, there exists an open ball B(x, r) lying inside O. Then, τ d = {O X : O is open} {φ} is a topology on X. τ d is called the topology induced by the metric d.

22 Example of I Definition Let τ 1 and τ 2 are two topologies on the set X. τ 2 is said to strictly stronger or stronger than τ 1 if τ 2 properly contain τ 1 or τ 2 τ 1 respectively. Definition Similarly, τ 2 is said to strictly weaker or weaker than τ 1 if τ 1 properly contain τ 2 or τ 2 τ 1 respectively. And both topologies are said to be comparable if τ 2 τ 1 and τ 1 τ 2.

23 Example of I Theorem In a topological space (X, τ), an arbitrary union of open sets and a finite intersection of open sets is open. Proof. We can proof this directly by the definition of topological space.

24 Example of II Theorem In a topological space (X, τ), an arbitrary intersection of closed sets and a finite union of closed sets is open. Proof:

25 Example of III Let F i be subset of X for all i in N and G = i=1 F i and H = n i=1 F i. Now F i is closed in X (X F i ) is open (X F i ) and i=1 X n (X F i ) are open sets i=1 F i and X i=1 F i and i=1 Hence G and H are closed sets. n F i are open sets i=1 n F i are closed sets i=1

26 Example of IV Theorem The union of infinite collection of closed sets in a topological space is not necessarily closed. Proof: n Let (R, τ) be topological space and F n = [0, n+1 ] for all n in N be subset of R. Here F n is closed. And F i = i=1 [ 0, 1 ] [ 0, 2 ]... [0, 1) 2 3 = [0, 1) closed set Therefore, i=1 F i is not closed set, even each F n is a closed set.

27 I Example of List all the possible topologies on X = {a, b}.

28 II Example of Let X = {a, b, c, d, e}. Determine whether or not each of the following classes of subsets of the X is a topology on X. 1 τ1 = {X, φ, {a}, {a, b}, {a, c}}. 2 τ2 = {X, φ, {a, b, c}, {a, b, d}, {a, b, c, d}}. 3 τ3 = {X, φ, {a}, {a, b}, {a, c, d}, {a, b, c, d}}.

29 III Example of Prove that (R 2, τ) is a topological space where the elements of τ are φ and the complements of finite sets of lines and points.

30 IV Example of Let τ = {R 2, φ} {G k : k R} be the class of the subsets of the plane R 2 where G k = {(x, y): x, y R, x > y + k} 1 Prove that τ is topology on R 2. 2 If τ is topology on R 2 if k R is replaced by k N, by k Q.

31 V Example of Let τ be the class consists of R, φ all infinite interval A q = (q, ) with q Q, the rationales. Show that τ is not topology on R.

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