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Unformatted text preview: 1 Point-Set Topology 1.1 Sufficient Conditions The following are sufficient conditions to guarantee each property. Normal. • X is metrizable. • X is compact Hausdorff. • X is regular and 2 nd countable. • X is regular and Lindel¨ of. Regular. • X is a subspace of a regular space. • X is a product of regular spaces. Metrizable. • X is regular and 2 nd countable. (Urysohn Metrization Theorem) • ( X is not metrizable if f ( x n ) → f ( x ), but x n 9 x where f is continuous.) Compact. • X is metrizable and (limit point compact or sequentially compact). • X is the product of compact spaces. • X is a closed subspace of a compact space. • X is the continuous image of a compact space. 1 st Countable. • X is metrizable. • X is a subspace of a 1 st countable space. • X is a countable product of 1 st countable spaces. 2 nd Countable. • X is a subspace of a 2 nd countable space. • X is a countable product of 2 nd countable spaces. 1.2 Theorems Sequence Lemma. Let X be a topological space; let A ⊂ X . If there is a sequence of points of A converging to x , then x ∈ A ; the converse holds if X is 1 St countable (or metrizable). Intermediate Value Theorem. Let f : X → Y be a continuous map, where X is a connected space and Y is an ordered set in the order topology. If a and b are two points of X and if r is a point of Y lying between f ( a ) and f ( b ), then there exists a point c of X such that f ( c ) = r . Extreme Value Theorem. Let f : X → Y be continuous, where Y is an ordered set in the order topology. If X is compact, then there exist points c and d in X such that f ( c ) ≤ f ( x ) ≤ f ( d ) for every x ∈ X . Lebesgue Number Lemma. Let A be an open covering of the metric space ( X,d ). If X is compact, there is a δ > 0 such that for each subset of X having a diameter less than δ , there exists an element of A containing it. Urysohn Lemma. Let X be a normal space; let A and B be disjoint closed subsets of X . Let [ a,b ] be a closed interval in the real line. Then there exists a continuous map f : X → [ a,b ] such that f ( x ) = a for every x in A , and f ( x ) = b for every x in B . Urysohn Metrization Theorem. Every regular space X with a countable basis is metrizable. Tietze Extension Theorem. Let X be a normal space; let A be a closed subspace of X . 1. Any continuous map of A into the closed interval [ a,b ] of R may be extended to a continuous map of all of X into [ a,b ]. 2. Any continuous map of A into R may be extended to a continuous map of all of X into R . 1 2 Homotopy Theory and Fundamental Groups 2.1 Essential Theorems Criterion for Homotopy Equivalence of CW-complexes. If ( X,A ) is a CW pair consisting of a CW complex X and a contractible subcomplex A , then the quotient map X → X/A is a homotopy equivalence....
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