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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.
7
votes
2
answers
441
views
Is every rational sequence topology homeomorphic?
Crossposted from Math.SE 4698387.
In the rational sequence topology, rationals are discrete and irrationals have a local base defined by choosing a Euclidean-converging sequence of rationals and decl …
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1
vote
Connected space being not locally connected at each point
The closed topologist's sine curve is the classic example of a connected but not locally connected space. But local neighborhoods away from the y-axis are copies of $\mathbb R$, which is connected.
To …
- 933
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 …
- 933
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 …
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0
votes
Accepted
Rothberger property and semi-open sets
First let's identify the semi-open sets. We first note all open sets are open. A simpler definition of the open sets are (assuming $p=0$),
$$\tau=\{U\subseteq\mathbb R:0\in U\Rightarrow\mathbb R\setmi …
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4
votes
2
answers
220
views
Must US extremally disconnected spaces be sequentially discrete?
Based upon discussion at Math.SE
Consider the property extremally disconnected, for which the closure of any open set remains open.
Frequently, this property is paired with the assumption of Hausdorff …
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1
vote
Must US extremally disconnected spaces be sequentially discrete?
KP answered the question, and in fact provided the answer to a stronger question: his space is strongly KC, showing that Hausdorff is quite necessary to show extremally disconnected spaces are both to …
- 933
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% …
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5
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 …
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1
vote
Hereditarily locally connected spaces
Let $X$ be a set, and let $\kappa$ be an infinite cardinal. Say sets in $X$ are closed if their cardinality is at most $\kappa$. (This class includes discrete spaces as you mentioned as well as spaces …
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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 …
- 933
4
votes
2
answers
128
views
Does there exist a non-hemicompact regular space for which the 2nd player in the $K$-Rothber...
Assume spaces are regular.
A space is $\sigma$-compact if and only if the second player in the Menger game has a winning Markov strategy (relying on only the most recent move of the opponent and the r …
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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 …
- 933
4
votes
1
answer
117
views
Idempotent relations on the unit square with closed graphs
A colleague and I are interested in idempotent relations from $I=[0,1]$ to $I$ - relations such that $R\circ R(x)=R(x)$ for all $x\in I$. Specifically, the graphs of the relations we care about must b …
- 933
2
votes
1
answer
139
views
Uniquely selecting points from open pairwise disjoint refinements of an open cover
Let $\mathcal U$ be an open cover of some space $X$.
Let $\{\mathcal V_\alpha:\alpha<\kappa\}$ enumerate all of its pairwise-disjoint
open refinements.
When is it possible to define sets $Z_\alpha$ su …
- 933