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Topological Space 1 2

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Contents

1. Topological Space 1

2. Subspace 2

3. Continuous Functions 2

4. Base 2

5. Separation Axiom 3

6. Compact Spaces and Locally Compact Spaces 3

7. Connected and Path connected 4

8. Product topology 5

9. Quotient Topology 5

1. Topological Space

Let X be a nonempty set and P(X) be the set of all subsets of X. A family T ⊂ P(X) is a topology on X if

(1) ∅, X ∈ T

(2) any union of members of T is a member of T

(3) any finite intersection of members of T is a member of T .

Elements of T are called open sets of X. A topological space is a pair (X, T ) where X is a nonempty set and T is a topology on X. A topological space (X, T ) is simply denoted by X when the topology T is specified. If U is an open set of X, we also say U is open in X.

A subset F is closed in X if X \ F is open in X.

Proposition 1.1. Let X be a topological space.

(1) ∅ and X are both closed sets.

(2) Any finite union of closed sets in X is closed.

(3) Any intersection of closed sets in X is closed.

Let X be a topological space. If U is an open set of X containing a point x ∈ X, U is called an open neighborhood of x. Let A be a subset of X and x ∈ A. We say that x is an interior point of A if there exists an open set U ⊂ A so that U is an open neighborhood of x. The set of interior points is denoted by A.

Proposition 1.2. A subset A of X is open if and only if A = A. Moreover, A is the largest open set contained in A.

A point x ∈ X is an adherent point of a subset A of X if the intersection of A and any open neighborhood of x is nonempty. The closure A of A is the set of all adherent points of A.

Lemma 1.1. A is closed if and only if A = A. A is the smallest closed set containing A; it is the intersection of all closed sets containing A.

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A point x ∈ X is a boundary point of A ⊂ X if it is an adherent point of both of A and X \ A. The topological boundary ∂A consists of all boundary points of A.

Theorem 1.1. A is the union of A and ∂A.

2. Subspace

Let X be a topological space and Y be a nonempty subset of X. Let TY be the subset of Y of the forms U ∩ Y for all U open in X.

Lemma 2.1. The family TY of subsets of Y forms a topology on Y.

The topology TY defined above is called the subspace topology of Y induced from X.

The pair (Y, TY) or the symbol Y is called a topological subspace of X. Elements of TY are also called relative open subsets of Y.

3. Continuous Functions

Let X and Y be topological spaces. A function f : X → Y is continuous if and only if f−1(V ) is open in X whenever V is open in Y. A function f : X → Y is called continuous at x if for every open neighborhood of f (x), there exists an open neighborhood U of x such that f (U ) ⊂ V.

Proposition 3.1. A function f : X → Y is continuous if and only if it is continuous at every point of X.

Proposition 3.2. Let f : X → Y and g : Y → Z be continuous functions. Then g ◦ f : X → Z is also continuous.

A bijection h : X → Y between spaces is a homeomorphism if both h, h−1 are continuous.

We say that X is homemorphic to Y or topologically equivalent to Y if there exists a homeomorphism h from X onto Y.

Lemma 3.1. Topological equivalence is an equivalence relation.

4. Base

Let X be a topological space. A family B of open sets of X is a base for the topology of X is every open set of X is a union of members of B.

Theorem 4.1. Let B be a family of open sets of X. The followings are equivalent.

(1) B is a base for the topology of X.

(2) For each x ∈ X and each open neighborhood U of x, there exists V ∈ B such that V is an open neighborhood of X.

Not every family of subsets of X can be a base for a topology. The following theorem is a characterization of a base for a topology on X.

Theorem 4.2. Let X be a nonempty set and B be a family of subsets of X. Then B is a base for a topology on X if and only if the following two properties holds:

(1) each x lies in at least member of B,

(2) if U, V ∈ B, x ∈ U ∩ V, there exists W ∈ B such that W ⊂ U ∩ V.

Definition 4.1. A topological space X is second countable if there is a countable family of open sets that forms a base for the topology for X.

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An open cover of a topological space X is a family of open sets U = {Uα: α ∈ Λ} so that S

αUα = X. An open subcover V of U is a subset of U such V is also an open cover of X.

Theorem 4.3. Let X be a second countable topological space. Then every open cover of X has a countable subcover.

We say that a subset A of X is dense if A = X. A topological space X is separable if there is a subset S that is countable and dense in X.

Theorem 4.4. If X is a second countable topological space, then X is separable.

5. Separation Axiom Let X be a topological space.

(1) X is T1 if for any x, y ∈ X with x 6= y, there exists an open neighborhood U of y such that x 6∈ U.

(2) X is T2 or Hausdorff if for any x, y ∈ X with x 6= y, there exist an open neighborhood U of x and an open neighborhood V of x such that U ∩ V = ∅.

(3) X is regular if for each closed subset E of X and each x ∈ X \ E, there exist disjoint open sets U, V such that x ∈ U and y ∈ V.

(4) X is T3 if X is a regular T1-space.

(5) X is normal if for each pari E and F of disjoint closed sets of X, there exist disjoint open sets U, V of X such that E ⊂ U and V ⊂ F.

(6) A T4-space is a normal T1-space.

Lemma 5.1.

T4=⇒ T3 =⇒ T2 =⇒ T1. Theorem 5.1. Every metric space is a T4-space.

Lemma 5.2. X is normal if and only if for each closed subset E of X and each open set W containing E, there exists an open set U containing E such that U ⊂ W.

Theorem 5.2. Let E and F be disjoint closed subsets of a normal space X. There exists a continuous function f : X → [0, 1] such that f = 0 on E and f = 1 on F.

Theorem 5.3. Let X be a normal space and Y be a closed subset of X. Suppose f is a bounded continuous real-valued function on Y. There exists a bounded continuous real- valued function F on X such that F = f on Y.

6. Compact Spaces and Locally Compact Spaces

A space is compact if every open cover has finite subcover. A subset K of a space X is a compact subset if it is a compact space with the subspace topology induced from X.

Proposition 6.1. Any finite union of compact subsets of a space is compact.

Proposition 6.2. A closed subspace of a compact space is compact.

Lemma 6.1. Let S be a compact subset of a space X. For each x ∈ X \ S, there exist disjoint open neighborhood U of x and V of S.

Corollary 6.1. A compact subset of a Hausdorff space is closed.

Theorem 6.1. A compact Hausdorff space is normal.

Theorem 6.2. Let f : X → Y be a continuous function from a compact space X to a space Y. Then f (X) is a compact subset of Y.

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Theorem 6.3. Let f be a continuous function from a compact space X to a Hausdorff space. If f is injective, then f is a homeomorphism of X and f (X).

A space X is locally compact if for each p ∈ X, there is an open neighborhood W of p such that W is compact.

Example 6.1. Let Rn be equipped with the standard Euclidean topology. Then Rn is locally compact Hausdorff space.

Theorem 6.4. Let X be a locally compact Hausdorff space and Y be a set consisting of X and one other element. Then there exists a unique topology for Y such that Y becomes compact Hausdorff space and the relative topology for X inherited from Y coincides with the original topology for X.

Definition 6.1. The space Y constructed in the Theorem 6.4 is called the one point com- pactification of X.

7. Connected and Path connected A space is disconnected if there exist open sets U, V such that

(1) X = U ∪ V (2) U ∩ V = ∅

(3) U 6= ∅, and V 6= ∅.

Proposition 7.1. The only nonempty closed and open subset of a connected space is the space itself.

A subset of a space is a connected subset if it is connected in the subspace topology.

Theorem 7.1. Let f : X → Y be a continuous function from a connected space into a space. Then f (X) is connected.

Theorem 7.2. Let {Eα} be a family of connected subsets of a space X such that Eα∩Eβ 6=

∅ for each α, β. ThenS

αEα is connected.

Let X be a space and x ∈ X. The connected component C(x) of x is the maximal connected subset of X containing x.

Proposition 7.2. Two connected components of X either coincide or are disjoint. The connected components of X form a partition of X.

A path in X from x0 to x1 is a continuous function γ : [0, 1] → X

such that γ(0) = x0 and γ(1) = x1. We also say that x0 and x1 is connected by γ. The space X is connected if any of its two points can be connected by a path.

Lemma 7.1. The relation “there is a path in X from x to y” is an equivalence relation on X.

The equivalent classes corresponding to the above equivalence relation are called path components of X.

Theorem 7.3. A path connected space is connected.

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8. Product topology

Let {Xα : α ∈ Λ} be a collection of spaces and X be product of sets Xα, i.e. X = Q

α∈ΛXα. Denote πα : X → Xα by (xα) 7→ xα called the projection from X to Xα. The product topology for X is the smallest topology for each each projection πα is continuous.

In this section, X always stands forQ

αXα with the product topology.

Theorem 8.1. Let E be a space and f : E → X be a function. Then f is continuous if and only if π ◦ f is continuous for all α.

Theorem 8.2. (Tychonoff’s Theorem )Any product of compact spaces is compact.

Now, let us consider finite product spaces. Suppose X1, · · · , Xn are spaces.

Theorem 8.3. The projections πj :Qn

j=1Xj → Xj are open mapping for all j.

Theorem 8.4. If X1, · · · , Xn are Hausdorff space, then Qn

i=1Xi is Hausdorff.

Theorem 8.5. If X1, · · · , Xn are path connected space, then Qn

i=1Xi is path connected.

Theorem 8.6. If X1, · · · , Xn are connected space, thenQn

i=1Xi is connected..

9. Quotient Topology

Let X be a topological space and R be an equivalence relation on X. The R-equivalence class of x ∈ X is denoted by [x]. We denote by X/R the set of all R-equivalence classes of elements of X. The function π : X → X/R sending an element x to its equivalence class [x]

is called the quotient map. Let T |X/R be the family of subsets U of X/R such that π−1(U ) is open in X.

Lemma 9.1. TX/R is a topology on X/R. The set X together with TX/R is called the quotient space of X. In this case, TX/R is called the quotient topology of X with respect to R. It is the largest topology such that π : X → X/R is continuous.

Theorem 9.1. Let Y be a topological space, X, R, X/R, π be as above. A function f : X/R → Y is continuous if and only if f ◦ π is continuous.

Theorem 9.2. Let f : X → Y be a continuous function. Let R be an equivalence relation on X such that f is constant on each equivalent class. Then there exists a continuous function g : X/R → Y such that f = g ◦ π.

Theorem 9.3. Let X, Y be compact Hausdorff space and f : X → Y be a continuous surjection. Define a relation R as follows: We say x1Rx2 if f (x1) = f (x2). Then R is an equivalence relation. Equip X/R with the quotient topology. Then X/R is homeomorphic to Y.

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