Compactness extends local stuff to global stuff because it's easy to make something satisfy finitely many restraints- this is good for bounds. Connectedness relies on the fact that ``clopen'' properties should …

Oct 3, 2021 · The wiki definiton defines a compactness of an interval as closed and bounded. In mathematics, specifically general topology, compactness is a property that generalizes the notion of …

Sep 7, 2013 · Compactness is useful even when it emerges as a property of subspaces: 3) Most of topological groups we face in math every day are locally compact, e.g $\mathbb {R}$, $\mathbb {C}$, …

Apr 25, 2013 · Also when trying to disprove compactness the books I've read start presenting strange covers that I would have never thought about. I think my real problem is that I didn't yet get the …

Feb 16, 2020 · Hence a key observation concerning compactness is that it is a non-trivial generalisation of finiteness. In Edwin Hewitt's Essay, "The rôle of compactness in analysis" he says that: "The …

Jun 15, 2019 · One notational hazard to watch out for: "closed" often means two totally different things simultaneously in this area due to conflicting historical naming conventions. A "closed manifold" is a …

The definition of compactness is that for all open covers, there exists a finite subcover. If you want to prove compactness for the interval [0,1] [0, 1], one way is to use the Heine-Borel Theorem that …

I'm looking for two examples: A space which is compact but not sequentially compact A space which is sequentially compact but not compact Explanations why the spaces are compact / not compact and

The latter is always implied by compactness, so that for sequential spaces we have compact implies sequentially compact (but not reversely, as $\omega_1$ in the order topology shows). Another …

Jun 13, 2016 · Compactness and sequential compactness in metric spaces Ask Question Asked 11 years, 6 months ago Modified 6 years, 11 months ago