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Results tagged with gn.general-topology
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user 73785
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
12
votes
Accepted
Pixley and Roy article request
I was visiting Auburn today and obtained a scan.
https://github.com/StevenClontz/research/blob/master/miscellaneous/SKM_C650i23042612550.pdf
12
votes
Compactly generated and paracompact $\Rightarrow$ Hausdorff?
Questions like these are easily answered with a search of pi-Base:
π-Base, Search for $k_3$-space+Paracompact+~$T_2$
Six counterexamples are listed there today, including Tyrone's example. I'll sugges …
10
votes
Is there a natural topology for sets of topological spaces?
There is no topology on the set of all [compact] topological spaces, because there is no set of all [compact] topological spaces.
Given a set of topological spaces, consider the power set of its union …
7
votes
Is a Hausdorff separable topological space that is uniform and complete necessarily a Polish...
Questions like these are often answerable by a search of the pi-Base (noting that every Hausdorff paracompact space is completely uniformizable): https://topology.pi-base.org/spaces?q=%20hausdorff%2B% …
6
votes
"All retracts are closed" and "all compacts are closed"
EDIT: This answer relied on an accepted answer elsewhere that has now been updated to remove an oversight. See my note below.
First I need to prove that the Arens-Fort space $X$ is not compactly gener …
5
votes
Accepted
Is the Fortissimo space on discrete $\omega_1$ radial?
Summarizing comments as an answer.
The standard interpretation of radial does not require that $\alpha\mapsto a_\alpha$ be continuous. Therefore, the Fortissimo space is radial, despite the lack of no …
5
votes
Countable chain condition in topology
Spearable implies ccc is theorem T21 of pi-Base, which references Counterexamples. The book doesn't provide a proof, but the result is standard. (Take a collection of pairwise disjoint open sets, if …
4
votes
"All retracts are closed" as separation axiom
Let $X$ be the rationals with their subspace topology, and $X^+=X\cup\{\infty\}$ be its one-point compactification.
Because $X$ is not locally compact, $X^+$ is not $T_2$.
The space $X^+$ has the prop …
3
votes
Accepted
A closed subset of a Dedekind-complete order has subspace topology equal to order topology
While I agree that it's pretty direct to show, I was unable to find a reference for a proof of this fact myself (I thought it was in Willard, but I thumbed through my copy and failed to find it). So h …
3
votes
Accepted
Is the class of rc-spaces closed under products?
Take $X$ to be an RC space which isn't $T_2$ such as the one-point compactification of the rationals. We will show $X^2$ is not RC. Note that it is not $T_2$ as its factors are not $T_2$.
First we wil …
3
votes
On the Menger property and the Alexandroff duplicate
The reference is here, provided you cannot find anything more classical.
Every closed subset of a Menger set is Menger. Thus if $A(X)$ is Menger, then its closed subset $X\times\{0\}\cong X$ is Menger …
3
votes
What are the names of the following classes of topological spaces?
For (1), $\omega$ bounded.
I'm unaware of names for the other concepts.
2
votes
Compactness of symmetric power of a compact space
Here's a direct proof (although showing your metric yields the topology of $\mathcal X/\sim$ is more useful). In a metric space, compactness is equivalent to sequential compactness: every infinite seq …
2
votes
Does there exist a non-hemicompact regular space for which the 2nd player in the $K$-Rothber...
Writing up a direct proof for Rothberger based upon Caruvana's references.
It's unclear why I didn't think to try it, but it's much easier to think about the K-Rothberger game in terms of its dual - b …
1
vote
Accepted
Idempotent relations on the unit square with closed graphs
https://www.researchgate.net/publication/281110530_Destruction_of_metrizability_in_generalized_inverse_limits
We worked out the details to get what we needed in that paper. Specifically, if $f$ is an …