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TOPOLOGY Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. Topological Space Subspace Continuous Functions Base Separation Axiom Compact Spaces and Locally Compact Spaces Connected and Path connected Product topology Quotient Topology 1 2 2 2 3 3 4 5 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. 1 2 CALCULUS 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. CALCULUS 3 S An open cover of a topological space X is a family of open sets U = {Uα : α ∈ Λ} so that α Uα = 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 realvalued 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. 4 CALCULUS 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 compactification 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 {ES α } be a family of connected subsets of a space X such that Eα ∩Eβ 6= ∅ for each α, β. Then α Eα 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. CALCULUS 5 8. Product topology Q Let {Xα : α ∈ Λ} be a collection of spaces and X be product of sets Xα , i.e. X = α∈Λ Xα . Denote πα : X → Xα by (xα ) 7→ xα called the projection from X to Xα . The product topology for X is the smallest Q topology for each each projection πα is continuous. In this section, X always stands for α Xα 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. Q Theorem 8.3. The projections πj : nj=1 Xj → Xj are open mapping for all j. Q Theorem 8.4. If X1 , · · · , Xn are Hausdorff space, then ni=1 Xi is Hausdorff. Q Theorem 8.5. If X1 , · · · , Xn are path connected space, then ni=1 Xi is path connected. Q Theorem 8.6. If X1 , · · · , Xn are connected space, then ni=1 Xi 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 x1 Rx2 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.