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Tag Archives: Hausdorff
Topology: Separation Axioms
Motivation The separation axioms attempt to answer the following. Question. Given a topological space X, how far is it from being metrisable? We had a hint earlier: all metric spaces are Hausdorff, i.e. distinct points can be separated by two … Continue reading
Posted in Notes
Tagged advanced, Hausdorff, metrisable topology, normal, regular, separation axioms, T1, T2, T3, T4, topology, urysohn's lemma, urysohn's metrisation theorem
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Topology: OnePoint Compactification and Locally Compact Spaces
Compactifications There’re lots of similarities between completeness and compactness, beyond the superficial resemblance of the words. For example, a closed subset of a compact (resp. complete) space is also compact (resp. complete). Two differences though: compactness is a topological concept … Continue reading
Topology: Hausdorff Spaces and Dense Subsets
Hausdorff Spaces Recall that we’d like a condition on a topological space X such that if a sequence converges, its limit is unique. A sufficient condition is given by the following: Definition. A topological space X is said to be Hausdorff if … Continue reading
Posted in Notes
Tagged advanced, continuity, dense subsets, Hausdorff, metric spaces, nets, topology
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Topology: Limits and Convergence
Following what we did for real analysis, we have the following definition of limits. Definition of Limits. Let X, Y be topological spaces and . If f : X{a} → Y is a function, then we write if the function: is … Continue reading
Posted in Notes
Tagged advanced, continuity, convergence, extended reals, Hausdorff, limits, sequences, topology
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