REVIEW OF POINT-SET TOPOLOGY I WOMP 2006 The
... Definition 4.4. A topological space X is called Hausdorff if for every x, y ∈ X, there exist open sets U and V such that x ∈ U , y ∈ V , and U ∩ V = ∅. Proposition 4.5. All closed subsets of a compact space are compact. The converse holds if the space is also Hausdorff. Proposition 4.6. Let f : X → ...
... Definition 4.4. A topological space X is called Hausdorff if for every x, y ∈ X, there exist open sets U and V such that x ∈ U , y ∈ V , and U ∩ V = ∅. Proposition 4.5. All closed subsets of a compact space are compact. The converse holds if the space is also Hausdorff. Proposition 4.6. Let f : X → ...
THE INTERSECTION OF TOPOLOGICAL AND METRIC SPACES
... Theorem 4.2. Every second countable regular space is a topological space. Proof. We will prove this theorem by imbedding an arbitrary second countable regular space X inside a space that we have shown to be metrizable. The space in which we imbed X is Rω with the product topology. Our ultimate goal ...
... Theorem 4.2. Every second countable regular space is a topological space. Proof. We will prove this theorem by imbedding an arbitrary second countable regular space X inside a space that we have shown to be metrizable. The space in which we imbed X is Rω with the product topology. Our ultimate goal ...
Building new topological spaces through canonical maps
... but it’s hard to picture what this looks like for n ≥ 3. (For n = 3, you can picture the 3-ball D3 sitting in R3 , but then how would you gather up the boundary sphere and glue it all to a single point? The resulting 3-sphere S 3 lives naturally in four dimensions, which isn’t so easy to imagine!) A ...
... but it’s hard to picture what this looks like for n ≥ 3. (For n = 3, you can picture the 3-ball D3 sitting in R3 , but then how would you gather up the boundary sphere and glue it all to a single point? The resulting 3-sphere S 3 lives naturally in four dimensions, which isn’t so easy to imagine!) A ...
REVIEW OF GENERAL TOPOLOGY I WOMP 2007 1. Basic Definitions
... Example 2.2. Consider the closed interval [0, 1] as a subset of R under its usual topology. Then [1, 0], (a, 1], and [0, b), for 0 < a, b < 1, are all open in [0, 1] under the subspace topology, being the intersections of [0, 1] with, respectively, the open subsets R, (a, ∞), and (−∞, b) of R. Hence ...
... Example 2.2. Consider the closed interval [0, 1] as a subset of R under its usual topology. Then [1, 0], (a, 1], and [0, b), for 0 < a, b < 1, are all open in [0, 1] under the subspace topology, being the intersections of [0, 1] with, respectively, the open subsets R, (a, ∞), and (−∞, b) of R. Hence ...
Homework 5 Solutions III.8 - University of South Alabama
... open set Bλ so that Bλ ⊆ Aλ (we can do this by the definition of a base). Then the Bλ cover X . Moreover, there are only countably many Bλ , so in fact we have a countable open cover {Bi }i∈N . Now for each i choose one Ai so that Bi ⊆ Ai . Then the Ai form a countable subcover of X . III.11: Find a ...
... open set Bλ so that Bλ ⊆ Aλ (we can do this by the definition of a base). Then the Bλ cover X . Moreover, there are only countably many Bλ , so in fact we have a countable open cover {Bi }i∈N . Now for each i choose one Ai so that Bi ⊆ Ai . Then the Ai form a countable subcover of X . III.11: Find a ...
List 4 - Math KSU
... • For all A, B ⊆ X satisfying ClA ∩ B = ∅ = A ∩ ClB , there exist U, V ∈ TX such that A ⊆ U , B ⊆ V and U ∩ V = ∅. Hint: if X is completely normal, consider (ClA ∩ ClB)c ⊆ X . 11. Let (X, TX ) be a compact Hausdorff space. Show that X is metrizable if and only if X is second countable. 12. Let (X, T ...
... • For all A, B ⊆ X satisfying ClA ∩ B = ∅ = A ∩ ClB , there exist U, V ∈ TX such that A ⊆ U , B ⊆ V and U ∩ V = ∅. Hint: if X is completely normal, consider (ClA ∩ ClB)c ⊆ X . 11. Let (X, TX ) be a compact Hausdorff space. Show that X is metrizable if and only if X is second countable. 12. Let (X, T ...
ALGEBRA I PRELIM, DECEMBER 2013 (SEE THE NEXT PAGE
... (d) Would the result in (c) still be true if we added a 0 to the right-hand side of the above two sequences? Prove it or give a counterexample. 5) For each of the following, give an example if possible, and if not possible, briefly explain why not. All rings in this problem are assumed to be commuta ...
... (d) Would the result in (c) still be true if we added a 0 to the right-hand side of the above two sequences? Prove it or give a counterexample. 5) For each of the following, give an example if possible, and if not possible, briefly explain why not. All rings in this problem are assumed to be commuta ...
Algebraic Topology Introduction
... In the case that f is a bijection, and a set of vertices forms a simplex in K if and only if the image of the vertices form a simplex in L, then it can be shown that g : |K| → |L| is a homeomorphism, and is called a simplicial homeomorphism. As Munkres aptly observes, “In practice, specifying a poly ...
... In the case that f is a bijection, and a set of vertices forms a simplex in K if and only if the image of the vertices form a simplex in L, then it can be shown that g : |K| → |L| is a homeomorphism, and is called a simplicial homeomorphism. As Munkres aptly observes, “In practice, specifying a poly ...
