Defn: If X is an ordered set and a ∈ X, then the following are rays in X: (a, +∞) = {x | x > a}, (−∞, a) = {x | x < a}, [a, +∞) = {x | x ≥ a}, (−∞, a] = {x | x ≤ a}.

Lemma: The collection of all open rays is a subbasis for the order topology.

15: The Product Topology

Let T_{X} denote the topology on X and T_{Y} denote the topology on Y.

Defn: Let X and Y be topological Spaces. The product topology on X ×Y is the topology having as basis B = {U × V | U ∈ T_{X}, V ∈ T_{Y} }.

Thm 15.1: If B_{X} is a basis for the topology of X and B_{Y} is a basis for the topology of Y, then D = {U × V | U ∈ B_{X}, V ∈ B_{Y} } is a basis for the topology of X × Y.

Ex. 1: If R has the standard topology, the product topology on R×R is the standard topology on R^{2}.

Defn: Let π_{1} : X_{1} × X_{2} → X_{1}, π_{1}(x_{1}, x_{2}) = x_{1}. π_{1} is the projection of X_{1} × X_{2} onto the first component. Note: If U ⊂ X_{1}, then π_{1}^{−1}(U) = U × X_{2}. Thus if U is open in X_{1}, then π_{1}^{−1}(U) is open in X_{1} × X_{2} Note: π_{1}^{−1}(U) ∩ π_{2}^{−1}(V) = U × V Thm 15.2: The collection S = {π_{1}^{−1}(U) | U open inX}∪{π_{2}^{−1}(V) | V open inY } is a subbasis for the product topology on X × Y.

16. The Subspace Topology. Defn: Let (X, T ) be a topological space, Y ⊂ X. Then the subspace topology on Y is the set

T_{Y} = {U ∩ Y | U ∈ T }

(Y, T_{Y}) is a subspace of X.

Lemma 16.1: If B is a basis for the topology of X, then the set

B_{Y} = {B ∩ Y | B ∈ B}

is a basis for the subspace topology on Y.

Lemma 16.2: Let Y be a subspace of X. If U is open in Y and Y is open in X, then U is open in X.

Lemma 16.3: If A_{j} is a subspace of X_{j}, j = 1, 2, then the product topology on A_{1} × A_{2} is the same as the topology A_{1} × A_{2} inherits as a subspace of X_{1} × X_{2}.

Note: Suppose Y ⊂ X where X is an ordered set with the order topology. The order topology on Y need not be the same as the subspace topology on Y

Ex 1:

Defn: Suppose Y ⊂ X where X is an ordered set. Y is convex if for all a, b ∈ Y such that a < b, then (a, b) ⊂ Y Ex. 1: (1, 2) ∪ (3, 4) ⊂ R. Ex. 2: (1, 2) ∪ (3, 4) ⊂ (1, 2) ∪ (3, 9). Lemma 16.4: Let X is an ordered set with the order topology. Let Y be a convex subset of X. Then the order topology on Y is the same as the subspace topology on Y. HW p91: 3 (prove your answer).

17. Closed Sets and Limit Points Defn: The set A is closed iff X − A is open. Thm 17.1: X be a topological space if and only if the following conditions hold:

(1) ∅, X are closed.

(2) Arbitrary intersections of closed sets are closed.

(3) Finite unions of closed sets are closed.

Thm 17.2: Let Y be a subspace of X. Then a set A is closed in Y if and only if it equals the intersection of a closed set of X with Y.

Thm 17.3: Let Y be a subspace of X. If A is closed in Y and Y is closed in X, then A is closed in X.

Defn: The interior of A = Int A = A^{0} = ∪_{U^{o}pen⊂A}U

Defn: The closure of A = Cl A = \overline{A} = ∪_{A⊂F^{closed}}F

Thm 17.4: Let Y be a subspace of X, A ⊂ Y. Let \overline{A} denote the closure of A in X. Then the closure of A in Y equals \overline{A} ∩ Y.

Defn: A intersects B is A ∩ B ≠ emptyset

Thm 17.5: Let A be a subset of the topological space X.

(a) x ∈ \overline{A} if and only if (x ∈ U^{open} implies U ∩A ≠ ∅).

(b) x ∈ \overline{A} if and only if (x ∈ B where B is a basis element implies B ∩ A ≠ ∅). neighborhood Ex Defn: x ∈ X is a limit point of A iff U ∩A {x} ≠ emptyset for every open set U. Defn: A′ = the set of all limit points of A. Thm 17.6: \overline{A} = A ∪ A′. Cor 17.7: A closed if and only if A′ ⊂ A. Defn: x_{n} converges to a limit x if for every neighborhood U of x, there exists a positive integer N such that n ≥ N implies x_{n} ∈ U. Note: limit point of a set is not the same as limit of a sequence. Defn: X is Hausdorff space if for all x_{1}, x_{2} ∈ X such that x_{1} ≠ x_{2}, there exists neighborhoods U_{1} and U_{2} of x_{1} and x_{2}, respectively such that U_{1} ∩ U_{2} = ∅.

Thm 17.8: Every finite point set in a Hausdorff space X is closed.

Defn: X is T_{1} if every one point set is closed.

Thm 17.9: Let X by T_{1}, A ⊂ X. Then x is a limit point of A is and only if every neighborhood of x contains infinitely many points of A.

Thm 17.10: If X is Hausdorff, then a sequence of points of X convereges to at most one point of X. (IFF?)

\frac{1}{n} converges to in the finite complement topology.

Thm 17.11: If X has the order topology, then X is Hausdorff. The product of two Hausdorff spaces is Hausdorff. A subspace of a Hausdorff space is Hausdorff.

17. Continuous Functions Defn: f^{−1}(V) = {x | f(x) ∈ V }. Defn: f : X → Y is continuous iff for every V open in Y, f^{−1}(V) is open in X. Lemma: f continuous if and only if for every basis element B, f^{−1}(B) is open in X. Lemma: f continuous if and only if for every subbasis element S, f^{−1}(S) is open in X. Thm 18.1: Let f : X → Y. Then the following are equivalent:

(1) f is continuous.

(2) For every subset A of X, f(\overline{A}) ⊂ \overline{f(A)}.

(3) For every closed set B of Y, f^{−1}(B)isclosedinX.

(4) For each x ∈ X and each neighborhood V of f(x), there is a neighborhood U of x such that f(U) ⊂ V. Defn: f : X → Y is a homeomorphism iff f is a bijection and both f and f^{−1} is continuous. topological property imbedding Thm 18.2 (a.) (Constant function) The constant map f : X → Y, f(x) = y_{0} is continuous. (b.) (Inclusion) If A is a subspace of X, then the inclusion map f : A → X, f(a) = a is continuous. (c.) (Composition) If f : X → Y and g : Y → Z are continuous, then g ∘ f : X → Z is continuous. (d.) (Restricting the Domain) If f : X → Y is continuous and if A is a subspace of X, then the restricted function f|_{A} : A → Y, f|_{A}(a) = f(a) is continuous.

(e.) (Restricting or Expanding the Codomain) If f : X → Y is continuous and if Z is a subspace of Y containing the image set f(X) or if Y is a subspace of Z, then g : X → Z is continuous.

(f.) (Local formulation of continuity) If f : X → Y and X = ∪U_{α}, U_{α} open where f|_{U_{α}}U_{α} → Y is continuous, then f : X → Y is continuous.

Thm 18.3 (The pasting lemma)¿ Let X = A ∪ B where A, B are closed in X. Let f : A → Y and g : B → Y be continuous. If f(x) = g(x) for all x ∈ A ∩ B, then h : X → Y, h(x) =

\{ \begin{matrix}f(x) & x ∈ A \\ g(x) & x ∈ B \end{matrix} iscontinuous.

THm 18.4: Let f : A → X × Y be given by the equations f(a) = (f_{1}(a), f_{2}(a)) where f_{1} : A → X, f_{2} : A → X. Then f is continuous if and only if f_{1} and f_{2} are continuous.

19. The Product Topology.

Defn: Let J be an index set. Given a set X, a J-tuple of elements of X is a function x : J → X. The αth coordinate of x = x_{α} = x(α)