Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
Questions tagged [gn.general-topology]
4434 questions
54
votes
3 answers
If any open set is a countable union of balls, does it imply separability?
If a metric space is separable, then any open set is a countable union of balls. Is the converse statement true?
UPDATE1. It is a duplicate of the question…
Fedor Petrov
- 102,548
53
votes
4 answers
Are the rationals homeomorphic to any power of the rationals?
I asked myself, which spaces have the property that $X^2$ is homeomorphic to $X$. I started to look at some examples like $\mathbb{N}^2 \cong \mathbb{N}$, $\mathbb{R}^2\ncong \mathbb{R}, C^2\cong C$ (for the cantor set $C$). And then I got stuck,…
HenrikRüping
- 10,279
45
votes
2 answers
Continuous bijections vs. Homeomorphisms
This is motivated by an old question of Henno Brandsma.
Two topological spaces $X$ and $Y$ are said to be bijectively related, if there exist continuous bijections $f:X \to Y$ and $g:Y \to X$. Let´s denote by $br(X)$ the number of homeomorphism…
Ramiro de la Vega
- 11,463
39
votes
4 answers
Topological Characterisation of the real line.
What is a purely topological characterisation of the real line( standard topology)?
Suryateja
- 501
37
votes
5 answers
When factors may be cancelled in homeomorphic products?
It is easy to see that if $A\times B$ is homeomorphic to $A\times C$ for topological spaces $A$, $B$, $C$, then one may not conclude that $B$ and $C$ are homeomorphic (for example, take $C=B^2$, $A=B^{\infty}$). The question is: for which $A$ such…
Fedor Petrov
- 102,548
36
votes
2 answers
Can non-homeomorphic spaces have homeomorphic squares?
I an wondering if there are non-homeomorphic spaces $X$ and $Y$ such that $X^2$ is homeomorphic to $Y^2$.
Pedro Perez
- 531
36
votes
5 answers
Example of sequences with different limits for two norms
I was explaining to my students that if there is an inequality between two norms, then there is an inclusion between their spaces of convergent sequences, with matching limits. I then proceeded to show examples of such inequalities on the normed…
Julien Puydt
- 2,013
36
votes
2 answers
A question about connected subsets of $[0,1]^2$
If $S⊂[0,1]^2$ intersects every connected subset of $[0,1]^2$ with a full projection on the $x$-axis, must $S$ have a connected component with a full projection on the $y$-axis?
An equivalent form：
If $S⊂[0,1]^2$ intersects every connected subset of…
mathoverflow12345
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35
votes
13 answers
Examples of non-metrizable spaces
I want to know some examples of topological spaces which are not metrizable. Of course one can construct a lot of such spaces but what I am looking for really is spaces which are important in other areas of mathematics like analysis or algebra. I…
Sudip Paul
- 111
33
votes
6 answers
Is a topology determined by its convergent sequences?
Just a basic point-set topology question: clearly we can detect differences in topologies using convergent sequences, but is there an example of two distinct topologies on the same set which have the same convergent sequences?
Tony
- 543
33
votes
4 answers
Connectedness in the language of path-connectedness
Is there a topological space $(C,\tau_C)$ and two points $c_0\neq c_1\in C$ such that the following holds?
A space $(X,\tau)$ is connected if and only if for all $x,y\in X$ there is a continuous map $f:C\to X$ such that $f(c_0) = x$ and $f(c_1) =…
Dominic van der Zypen
- 45,374
33
votes
4 answers
Can a connected planar compactum minus a point be totally disconnected?
What the title said. In a slightly more leisurely fashion:-
Let $X$ be a compact, connected subset of $\mathbb{R}^2$ with more than one point, and let $x\in X$. Can $X\smallsetminus\{x\}$ be totally disconnected?
Note that the Knaster-Kuratowski…
HJRW
- 24,015
32
votes
2 answers
How do you axiomatize topology via nets?
Let $X$ be a set and let ${\mathcal N}$ be a collection of nets on $X.$
I've been told by several different people that ${\mathcal N}$ is the collection of convergent nets on $X$ with respect to some topology if and only if it satisfies some axioms.…
Fabrizio Polo
- 671
32
votes
3 answers
Which spaces are inverse limits of discrete spaces ?
There is the following theorem:
"A space $X$ is the inverse limit of a system of discrete finite spaces, if and only if $X$ is totally disconnected, compact and Hausdorff."
A finite discrete space is totally disconnected, compact and Hausdorff and…
HenrikRüping
- 10,279
32
votes
1 answer
Homeomorphisms and disjoint unions
Let $X$ and $Y$ be compact subsets of $\mathbb{R}^n$. Assume that $X \sqcup X \cong Y \sqcup Y$ (here $X \sqcup X$ is the disjoint union of two copies of $X$, considered as a topological space, and similarly for $Y \sqcup Y$). Then I'm pretty sure…
Sam
- 546