Problem 3, Page 100 Show that if A is closed in X and B is closed in
... Proof:Given U × V ⊂ B × D, where U is open in B and V is open in D. Then (f × g)−1 (U × V ) = {(a, c) ∈ A × C | f (a) ∈ U and g(c) ∈ V }. That is, (f × g)−1 (U × V ) = {(a, c) ∈ A × C | a ∈ f −1 (U ) and c ∈ g −1 (V )}. So (f × g)−1 (U × V ) = f −1 (U ) × g −1 (V ). Since both f and g are continuous ...
... Proof:Given U × V ⊂ B × D, where U is open in B and V is open in D. Then (f × g)−1 (U × V ) = {(a, c) ∈ A × C | f (a) ∈ U and g(c) ∈ V }. That is, (f × g)−1 (U × V ) = {(a, c) ∈ A × C | a ∈ f −1 (U ) and c ∈ g −1 (V )}. So (f × g)−1 (U × V ) = f −1 (U ) × g −1 (V ). Since both f and g are continuous ...
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... In the following, let X be a topological space. Theorem 1. Suppose Y ⊆ X is equipped with the subspace topology, and A ⊆ Y . Then A is closed in Y if and only if A = Y ∩ J for some closed set J ⊆ X. Proof. If A is closed in Y , then Y \ A is open in Y , and by the definition of the subspace topology ...
... In the following, let X be a topological space. Theorem 1. Suppose Y ⊆ X is equipped with the subspace topology, and A ⊆ Y . Then A is closed in Y if and only if A = Y ∩ J for some closed set J ⊆ X. Proof. If A is closed in Y , then Y \ A is open in Y , and by the definition of the subspace topology ...
p. 1 Math 490 Notes 8 Convergence, and Hausdorff and T1 Spaces
... In a topological space (X, τ ), we say a countable sequence of points {xn } converges (or τ converges) to a point x ∈ X iff for each τ -nbhd U of x, there exists a corresponding positive integer N such that xn ∈ U for all n ≥ N . In familiar spaces such as R and R2 with the usual topology, sequences ...
... In a topological space (X, τ ), we say a countable sequence of points {xn } converges (or τ converges) to a point x ∈ X iff for each τ -nbhd U of x, there exists a corresponding positive integer N such that xn ∈ U for all n ≥ N . In familiar spaces such as R and R2 with the usual topology, sequences ...
Tutorial 12 - School of Mathematics and Statistics, University of Sydney
... solutions. Replacing f by −f if need be, we may suppose that f (c) > f (a). The restriction of f to the compact interval [a, b] must achieve a maximum value M ≥ f (c) on [a, b]. Suppose that f (d) = M , where d ∈ (a, b). By the assumption about f there must be two solutions of f (x) = M + 1. Choose ...
... solutions. Replacing f by −f if need be, we may suppose that f (c) > f (a). The restriction of f to the compact interval [a, b] must achieve a maximum value M ≥ f (c) on [a, b]. Suppose that f (d) = M , where d ∈ (a, b). By the assumption about f there must be two solutions of f (x) = M + 1. Choose ...
21. Metric spaces (continued). Lemma: If d is a metric on X and A
... Thm 21.5: If X is a topological space, and if f, g : X → R are continuous, then f + g, f − g, f · g are continuous. If g(x) 6= 0 for all x, then f /g is continuous. Defn: Let fn : X → Y be a sequence of functions from the set X to the topological space Y . Then the sequence of functions (fn ) conver ...
... Thm 21.5: If X is a topological space, and if f, g : X → R are continuous, then f + g, f − g, f · g are continuous. If g(x) 6= 0 for all x, then f /g is continuous. Defn: Let fn : X → Y be a sequence of functions from the set X to the topological space Y . Then the sequence of functions (fn ) conver ...
Assignment 2 SOLUTION MATH 6540 (1) Show that a subset K of a
... it follows that the set A ∪ (0, 1] × {0} ∪ [0, 1) × {1} is closed. So it suffices to show that every point hx, ii ∈ (0, 1] × {0} ∪ [0, 1) × {1} is a limit point of A. First consider hx, 0i for x > 0. Since x > 0, any open interval around hx, 0i is of the form J = (ha, bi, hc, di) where a < x. If we ...
... it follows that the set A ∪ (0, 1] × {0} ∪ [0, 1) × {1} is closed. So it suffices to show that every point hx, ii ∈ (0, 1] × {0} ∪ [0, 1) × {1} is a limit point of A. First consider hx, 0i for x > 0. Since x > 0, any open interval around hx, 0i is of the form J = (ha, bi, hc, di) where a < x. If we ...
General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.