5
votes
Accepted
Is there a continuous bijective mapping of $R$ into a compactum
We can modify the example of the continuous bijection $[0, 1) \to S^1$ slightly; namely, thinking of $\mathbb{R}$ as an open interval $(0, 1)$, it admits a continuous bijection to a compact subspace ...
- 432k
2
votes
Closed sets definition in complex analysis
Neither is more correct since we can prove:
A set $A \subset \mathbb{C}$ contains all its limit points if and only if it contains all its boundary points.
However, the first definition is more ...
- 26.1k
2
votes
Examples of topological spaces that closed+boundness implies compactness
Here is a criterion similar to what you asked for in the question. If $(X,d)$ is a metric space, we say that $E\subseteq X$ is totally bounded if for every $\varepsilon>0$, there exists a finite ...
- 20.7k
2
votes
When will "Retract $\iff$ Deformation Retract" hold true?
Statement 1 is almost never true. For example if $A$ is a point it is always a retract of $X$ (via the unique map $X \to A$) but if it's a deformation retract then $X$ must be contractible. Being a ...
- 432k
1
vote
Examples of topological spaces that closed+boundness implies compactness
In locally convex theory, Montel's theorem was the reason to call barrelled spaces with the property that all closed bounded sets are compact Montel spaces. As a rule of thumb, almost all locally ...
- 12.3k
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