EXAM IN MA3002 GENERAL TOPOLOGY
... We give X/A the quotient topology. a) Show that if F ⊆ X is closed and F ∩ A = ∅, then π(F ) is closed in X/A. b) Show that if X/A is Hausdorff, then the set A is closed in X. Problem 9 In the following we let S 1 be the unit circle, and N be the natural numbers with the discrete topology. Let X = S ...
... We give X/A the quotient topology. a) Show that if F ⊆ X is closed and F ∩ A = ∅, then π(F ) is closed in X/A. b) Show that if X/A is Hausdorff, then the set A is closed in X. Problem 9 In the following we let S 1 be the unit circle, and N be the natural numbers with the discrete topology. Let X = S ...
Unified operation approach of generalized closed sets via
... Corollary 2.9 Let (X, τ, I) be a topological space and A and F subsets of X. If A is I-gclosed and F is closed in (X, τ ), then A ∩ F is I-g-closed. Proof. Since A ∩ F is closed in (A, τ |A), then A ∩ F is IA -g-closed in (A, τ |A, IA ). By Theorem 2.8, A ∩ F is I-g-closed. 2 Example 2.10 Corollary ...
... Corollary 2.9 Let (X, τ, I) be a topological space and A and F subsets of X. If A is I-gclosed and F is closed in (X, τ ), then A ∩ F is I-g-closed. Proof. Since A ∩ F is closed in (A, τ |A), then A ∩ F is IA -g-closed in (A, τ |A, IA ). By Theorem 2.8, A ∩ F is I-g-closed. 2 Example 2.10 Corollary ...
Chapter 1: Sets, Functions and Enumerability
... a) O (odd numbers) is enumerable. Use f(n) = 2n-1. b) {1, 2, 3} is enumerable. (But not enumerably infinite.) c) Any subset S of P is enumerable. Use f(n) = n, n ∈ S; undefined otherwise. d) φ is enumerable. We can use the partial function e, whose domain is empty: undefined everywhere! Then the ran ...
... a) O (odd numbers) is enumerable. Use f(n) = 2n-1. b) {1, 2, 3} is enumerable. (But not enumerably infinite.) c) Any subset S of P is enumerable. Use f(n) = n, n ∈ S; undefined otherwise. d) φ is enumerable. We can use the partial function e, whose domain is empty: undefined everywhere! Then the ran ...
Lecture 3
... a subspace of the space of all n × n matrices with real entries (the latter may be identified with R n ). If GL(n, R) were connected then so would be its image under a continuous map. Well, the determinant map d : GL(n, R) −→ R is continuous but the image is the real line minus the origin. The same ...
... a subspace of the space of all n × n matrices with real entries (the latter may be identified with R n ). If GL(n, R) were connected then so would be its image under a continuous map. Well, the determinant map d : GL(n, R) −→ R is continuous but the image is the real line minus the origin. The same ...
Locally compact groups and continuous logic
... for each x) on metric spaces with unbounded orbits. The first part of this section is devoted to some modifications of this property. We will show how continuous logic can work in these cases. In fact metric groups from these classes can be presented as reducts of continuous metric structures which ...
... for each x) on metric spaces with unbounded orbits. The first part of this section is devoted to some modifications of this property. We will show how continuous logic can work in these cases. In fact metric groups from these classes can be presented as reducts of continuous metric structures which ...
0,ω into continuous images of Valdivia compacta
... Proof of Theorem 1: (1) ⇒ (2) follows from the well known fact that [0, ω1 ] is not a Corson compact (see e.g. [DG]). (1) ⇒ (4) Let A be a dense Σ-subset of K and F ⊂ A an arbitrary relatively closed subset. Then F is ℵ1 -closed in K (by Lemma 1), so f (F ) is ℵ1 -closed in L by Lemma 4. But as L is ...
... Proof of Theorem 1: (1) ⇒ (2) follows from the well known fact that [0, ω1 ] is not a Corson compact (see e.g. [DG]). (1) ⇒ (4) Let A be a dense Σ-subset of K and F ⊂ A an arbitrary relatively closed subset. Then F is ℵ1 -closed in K (by Lemma 1), so f (F ) is ℵ1 -closed in L by Lemma 4. But as L is ...
