All Questions
14 questions
1
vote
0
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90
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Well-embedded type property for bounded functions
According to @Tyrone the term well-embedded set was first used in Measures on Metacompact Spaces by W. Moran.
In the article Extensions of Zero-sets and of Real-valued Functions by R. Blair and A. ...
- 1,201
15
votes
1
answer
796
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What is this equivalence relation on topological spaces: there are bijective continuous maps in both directions
Consider the following equivalence relation on topological spaces:
$X\sim Y$ $:\Longleftrightarrow$ there are bijective continuous maps $\phi:X\to Y$ and $\psi:Y\to X$.
Note that there are no ...
- 13.6k
1
vote
0
answers
84
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Is there a standard name for the following class of functions on non-Hausdorff manifolds?
Let $M$ be a (not necessarily Hausdorff) smooth manifold. Given an open chart $U\subset M$ and a compactly-supported smooth function $f:U\to\mathbb{R}$ on $U$, define $\widetilde{f}:M\to\mathbb{R}$ by ...
- 2,178
2
votes
1
answer
198
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A stronger version of paracompactness
Given a topological space $(X,\tau)$, recall that a cover $\mathcal{U}$ of $X$ is locally finite if for every point $x\in \mathcal{U}$ has a neighborhood $U$ that intersects finitely many elements of $...
- 775
1
vote
1
answer
204
views
Name of a space with both a topology and a metric that are not compatible?
Let $(X,\tau,d)$ be a space where $\tau$ is a topology and $d$ is a metric, where the topology $\tau$ is not necessarily compatible with $d$.
Is there a canonical name for such a structure (maybe ...
- 775
2
votes
2
answers
589
views
What to call a continuous function with preimage preserving nowhere-density?
Currently I am reading some basic literature on descriptive set theory and boolean algebras. And this comes out a lot, for example in results like:
Let $X$ and $Y$ be topological spaces, and $f:X \to ...
3
votes
0
answers
78
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Classification of limit points
Let $X$ be a subset of a topolgical space with no open points. Then
$$\overline{X}=X_1\sqcup X_2\sqcup X_3\sqcup X_4\sqcup X_5$$
where $X_1$ are isolated points of $X$,
$X_2$ are interior points, $X_3=...
- 6,891
3
votes
0
answers
74
views
Equivalence relation induced by Kolmogorov quotients
Recall: given a (possibly non-$T_0$) topological space $X$, its Kolmogorov quotient $KX$ is the $T_0$ topological space formed by $X/\sim$ where $x\sim y$ if they are topologically indistinguishable. ...
- 39k
11
votes
1
answer
355
views
Name for topological spaces where "every point has a local base wellordered by reverse inclusion"?
There are many properties regarding local bases of a topological space, like first countable if every point has a countable local base.
Is there a similar name for a space where "every point has a ...
- 775
6
votes
1
answer
186
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Reference request: A collection of topologies on $\mathbb{N}$ formed via series
First, some quick notation: for any series $\sum_{n=1}^\infty a_n$ whose terms are positive real numbers, and for any subset $M = \{m_1, m_2,...\} \subseteq \mathbb{N}$, we write $\sum_M a_n$ to mean ...
- 6,546
9
votes
0
answers
569
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A standard name for a function satisfying the intermediate value theorem?
Do you know any (standard) name for a function $f:\mathbb R\to\mathbb R$ having the following weak intermediate value property:
$(*)$ for any connected subset $C\subset \mathbb R$ and points $a,b\...
- 41.9k
9
votes
1
answer
322
views
What is the (genuine) name for the Gutik hedgehog?
Working with non-regular topological semigroups, my collegue Oleg Gutik discovered a special space $H$ which we named Gutik's hedgehog. It is homeomorphic to the space
$$H:=\{(0,0)\}\cup\{(\tfrac1n,0):...
- 41.9k
4
votes
5
answers
1k
views
A generalized diagonal?
A simple question. Let $ f:X\to Y $ be a function and let $ E_f:=\{(x, y): f (x)=f (y)\}\subset X\times X $. What is the name of the set $ E(f) $? It would be nice to have some reference also. It ...
9
votes
0
answers
685
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Name for a topological space where every closed set contains a closed point
A coauthor and I have stumbled upon a useful topological property -- namely, we are interested in the property that every nonempty closed set contains a closed point. However, neither of us are ...
- 1,802