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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
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
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). Two differences though: compactness is a topological concept … Continue reading
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
Posted in Notes
Tagged advanced, compact spaces, finite intersection property, four colour theorem, topology
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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
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
Posted in Notes
Tagged compact spaces, metric spaces, nets, sequences, sequentially compact spaces, subnets
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Mathematics and Such 