![important result of the fuzzy tychonoff theorem and](http://s1.studyres.com/store/data/019758158_1-7d1d952cd7d8d3a3048a86d2f0598c6a-300x300.png)
An Introduction to Topological Groups
... Example 2.8. In R with the Euclidean topology, the set [0, 1] is closed. This is because R \ [0, 1] = (−∞, 0) ∪ (1, ∞), which is the union of two open intervals. Example 2.9. In (X, P(X)) every subset of X is closed. This is the case because for any F ⊂ X we have X \ F ∈ P(X). Given a topological sp ...
... Example 2.8. In R with the Euclidean topology, the set [0, 1] is closed. This is because R \ [0, 1] = (−∞, 0) ∪ (1, ∞), which is the union of two open intervals. Example 2.9. In (X, P(X)) every subset of X is closed. This is the case because for any F ⊂ X we have X \ F ∈ P(X). Given a topological sp ...
CONVERGENCE Contents 1. Introduction
... Proposition 1.3. Let f : X → Y be a mapping between two metric spaces. TFAE: a) f is continuous. b) If xn → x in X, then f (xn ) → f (x) in Y . In other words, continuous functions between metric spaces are characterized as those which preserve limits of convergent sequences. Proposition 1.4. Let x ...
... Proposition 1.3. Let f : X → Y be a mapping between two metric spaces. TFAE: a) f is continuous. b) If xn → x in X, then f (xn ) → f (x) in Y . In other words, continuous functions between metric spaces are characterized as those which preserve limits of convergent sequences. Proposition 1.4. Let x ...