Tag Archives: compact spaces

Polynomials and Representations XXIX

Characters Definition. The character of a continuous G-module V is defined as: This is a continuous map since it is an example of a matrix coefficient. Clearly for any . The following are quite easy to show: The last equality, … Continue reading

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Polynomials and Representations XXVIII

Starting from this article, we will look at representations of . Now, itself is extremely complicated so we will only focus on representations of particular types. Generally, for any topological group G, we want: to be a continuous homomorphism of groups. Continuous … Continue reading

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Topology: One-Point 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). Also, the image of a compact (resp. complete) space under a … Continue reading

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Topology: Finite Intersection Property (Omake)

The whole point of this article is the following seemingly trivial observation. Theorem. A topological space X is compact if and only if it satisfies the finite intersection property (F.I.P.): if is a collection of closed subsets of X such that … Continue reading

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Topology: More on Compact Spaces

In the previous article, we defined compact spaces as those where every open cover has a finite subcover, i.e. if then we can find a finite set of indices such that On an intuitive level, one should imagine a compact … Continue reading

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Topology: Sequentially Compact Spaces and Compact Spaces

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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