Tag Archives: Hausdorff

Topology: Separation Axioms

Posted on March 15, 2013 by limsup

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 →

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Topology: One-Point Compactification and Locally Compact Spaces

Posted on March 3, 2013 by limsup

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 →

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Topology: Hausdorff Spaces and Dense Subsets

Posted on February 18, 2013 by limsup

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 →

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Topology: Limits and Convergence

Posted on February 2, 2013 by limsup

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 →

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