Modules vs Vector Spaces
Please go through this article once, it will clear your viewpoint about these two structures.
Compactness in Metric Spaces
The idea of compactness comes from the fact that these are the nice set next to the finite sets. i.e. It shares many properties of finite sets in a topological space. The first thing common between finite and compact sets is, these are closed and bounded. Every sequence has a limit point. A continuous image of a compact set is compact and many more. Compactness has a very close relation to completeness . Let X be a topological space then the definition of compactness is defined in the following way-: X is said to be compact if every open covering of X has a finite sub-covering. Heine-Borel Theorem -: Heine-Borel make our work easy for Euclidean spaces by generalizing that, ‘ K is a compact subset of R if and only if it is closed and bounded. ’ But above theorem doesn’t work well for all metric spaces , for example -: Take N with discrete metric. What is missing here? Completeness? No! Boundedness? No! Closedness? No! Then what? The thing that is missing is totally bou...
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