Tag Archives: nets

Topology: Sequentially Compact Spaces and Compact Spaces

Posted on February 24, 2013 by limsup

We’ve arrived at possibly the most confusing notion in topology/analysis. First, we wish to fulfil an earlier promise: to prove that if C is a closed and bounded subset of R and f : R → R is continuous, then f(C) is closed and bounded. [ As … 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: Nets and Points of Accumulation

Posted on February 5, 2013 by limsup

Recall that a sequence in a topological space X converges to a in X if the function f : N* → X which takes is continuous at . Unrolling the definition, it means that for any open subset U of X containing a, the set contains (N, ∞] for some N. In … Continue reading →

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