All Questions
Tagged with gn.general-topology terminology
20 questions with no upvoted or accepted answers
9
votes
0
answers
569
views
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
0
answers
685
views
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
6
votes
0
answers
108
views
How to call a point in a space having the property that there is essentially one $\omega$-sequence converging to it?
Consider the point $x=\langle \omega_1,\omega\rangle$ in the Tychonov plank $(\omega_1 + 1)\times(\omega + 1)$. Then there is essentially only one sequence (of length $\omega$) converging to it, ...
- 1,855
6
votes
0
answers
322
views
Terminology for notion dual to "support"
If $X$ is a set (feel free to think of it as finite, but it doesn't have to be) and $f$ a real function on $X$, call the support $\operatorname{supp} f$ the subset of $X$ consisting of all elements $i\...
- 21.5k
4
votes
0
answers
177
views
Continuity of equivalence relations
A function $\varphi : X \rightarrow Y$ between two topological spaces is continuous if and only if $\varphi(\,\overline{A}\,) \subset \overline{\varphi(A)}$ for all $A \subset X$.
This property can ...
- 18.7k
4
votes
0
answers
137
views
Does this property of scattered spaces have a name?
(Note: I asked this question at MSE a week ago and received no answer, so I am now reposting it here.)
Let $K$ be a (Hausdorff) scattered topological space and for each ordinal $\alpha$ denote by $K^{...
- 2,378
3
votes
0
answers
94
views
Pseudocompactness, countable compactness and locally finite open covers
Let $(P_1)$ be the property: Every locally finite open cover of $X$ has finite subcover.
Let $(P_2)$ be the property: Every locally finite open cover of $X$ is finite.
Let $(P_3)$ be the property: ...
- 1,201
3
votes
0
answers
78
views
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
132
views
Terminology for a generalization of the initial topology
This may be a simple piece of terminology, but I have not located it. For the initial topology, we are given a set of functions, indexed by $\alpha$, $f_{\alpha}:X\rightarrow Y_{\alpha}$, where each ...
- 1,422
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
3
votes
0
answers
78
views
Name for mappings that are "not quite projections"
Is there a known name for the following definition?
Consider topological spaces $X$, $Y$ and $f: X \rightarrow Y$ a continuous mapping. Then, $f$ is an "almost projection" if there is a topological ...
- 1,368
2
votes
0
answers
74
views
Is there a literature name for this concept of a "graded metric"?
Given a space $X$, I have been thinking about a function $d\colon X \times X \times \mathbb{N} \to \mathbb{R}_{\geq 0}$ (i.e. with values that are nonnegative reals) with the properties below. One may ...
- 21
2
votes
0
answers
103
views
$n$-connected spaces (terminology)
A graph is called $n$-connected if it remains connected after removal $\le n$ vertices.
Question. What is the name of an analogous property of topological spaces: a space that remains connected after ...
- 41.9k
2
votes
0
answers
146
views
How do you call a map which sends convergent sequences to pre-compact ones ?
In my work I encountered a map $f$ between two metric spaces $X$ and $Y$ that was not continuous (at least I couldn't prove it was), but I was able to prove that convergent sequences $(x_n)$ in $X$ ...
- 4,123
1
vote
0
answers
90
views
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
1
vote
0
answers
84
views
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
1
vote
0
answers
84
views
Terminology for the property: "Each uncountable disjoint open family is locally countable"
Suppose that a topological space $X$ satisfies the following property
(P): "Each uncountable disjoint open family is locally countable",
where a family $\mathcal U$ of subsets of $X$ is ...
- 505
0
votes
0
answers
42
views
Name for a sequence of open sets, each dense in the complement of the previous ones in the subspace topology
Let $X$ be a topological space. Let $\mathfrak{U} = \langle U_\alpha:\alpha\in\gamma\rangle$ be a sequence of non-empty open subsets of $X$ of length $\gamma$ ($\gamma$ an ordinal). Say (for now) that ...
- 1,855
0
votes
0
answers
41
views
Selectively countable Boolean algebras of sets (terminology)
I am interested in the name for the following property of a Boolean algebra $\mathcal A$ of subsets of a set $X$:
$(\star)$ for any sequence $(A_n)_{n\in\omega}$ of pairwise disjoint nonempty sets in $...
- 41.9k
0
votes
0
answers
139
views
Why the name 'regular' space?
It is well known that a regular space is a topological space $X$ with these two properties:
1)All one point sets are closed.
2)For every $x\in X$ and every closed set $B$ (such that $x\notin B$), ...
- 883