Questions tagged [gn.general-topology]
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
4,447
questions
4
votes
1
answer
220
views
K-analytic spaces whose any compact subset is countable
A regular topological space $X$ is called
$\bullet$ analytic if $X$ is a continuous image of a Polish space;
$\bullet$ $K$-analytic if $X$ is the image of a Polish space $P$ under an upper ...
- 40.9k
3
votes
0
answers
132
views
Is there a normal space with a $G_\delta$ diagonal which is not submetrizable?
A space has a $G_\delta$-diagonal if its diagonal can be written as the intersection of countably many open subsets of the square. A space is submetrizable if it has a weaker metrizable topology. ...
- 4,343
9
votes
0
answers
216
views
Point-free topology, but with $\sigma$-algebras instead of spaces
I have a question about $\sigma$-algebras in relation to point-free topology. The question was inspired by a comment on a similar question I had:
If abstract $\sigma$-algebras (i.e. certain boolean ...
- 60.2k
6
votes
0
answers
143
views
Countably compact non-compact perfect spaces
Recall that a space is countably compact if every infinite set has an accumulation point. A space is perfect if every closed set is a countable intersection of open sets. One of the classical ...
- 4,343
9
votes
4
answers
494
views
Infimum of a finite number of distances in the plane
Suppose one has a finite number of distances $d_1,\ldots,d_k$ on the Euclidean plane all of which metricize the usual Euclidean topology.
Define for each pair of points $x$ and $y$ in the plane
$$d(x,...
- 2,743
2
votes
1
answer
910
views
Proving that family of sets has non-empty intersection
Let's say I have an object which can be viewed as family of sets $\mathfrak{S} \subseteq 2^S$, and I want to prove that its intersection is non-empty. What is known already:
$S$ is set of measurable ...
- 105
1
vote
0
answers
65
views
Quantification over Nets
On a topological space $X$, a net is defined as a map $A \ni \alpha \longmapsto x_{\alpha} \in X$ from a directed set $A$.
With this, compactness of $X$ (for instance) is equivalent to "every net $(...
- 145
6
votes
0
answers
354
views
Topological Singularities in Affine Varieties
Let $X$ be an affine variety over $\mathbb{C}$. Let $x\in X$.
If $x$ is non-singular, then $x$ is locally holomorphic (in the Euclidean topology). See here for a relevant MO post.
By results of ...
- 8,394
2
votes
1
answer
101
views
Rothberger game and Meager in itself sets
On $(\mathbb{R}, \tau)$ the euclidean space of real numbers, we define a new topology by letting $\tau^{*}=\{X\subseteq \mathbb{R}: X=\emptyset \hspace{0.1cm}\mbox{or}\hspace{0.1cm}\mathbb{R}\setminus ...
- 11.9k
0
votes
1
answer
88
views
Morphism of schemes with non-sober image
Let $f:X\rightarrow Y$ be a morphism of schemes. Can the image of $f$ endowed with the subspace topology not be sober?
- 27.4k
2
votes
0
answers
208
views
A closed point in the closure of any point in the closure of any point of an irreducible scheme
Let $X$ be the underlying space of an irreducible scheme. In particular, $X$ is non-empty.
Note that the closure of any point is a closed irreducible subset. We say that a point has codimension $n$ ...
CommunityBot
- 1
4
votes
0
answers
101
views
How is the instanton Floer homology of Seifert fibrations related to that of a trivial fibration
My question centers around the relationship of the Chern-Simons theories of a Seifert fibration and the trivial product space $\Sigma_g \times S^1$, and its implication for instanton Floer homology. ...
CommunityBot
- 1
3
votes
1
answer
141
views
A reference for a (folklore?) characterization of K-analytic spaces
I am writing a paper on K-analytic spaces and need the following known characterization.
Theorem. For a regular topological space $X$ the following conditions are equivalent:
(1) $X$ is a continuous ...
CommunityBot
- 1
2
votes
1
answer
110
views
Is there a locally countable and weakly Lindelöf space which is not ccc
Is there a locally countable and weakly Lindelöf space which is not ccc?
A space $X$ is locally countable if for each point $x\in X$ there is an open neighbourhood $O_x$ of $x$ such that $|O_x| < \...
- 17.4k
8
votes
2
answers
911
views
When does Scott topology generated by specialization order induced by a sober space (X,$\tau$) equal the initial topology $\tau$?
Let X be a $T_{0}$ space. The specialization order ≤ on X is that if x is contained in cl{y}, then we call "x≤y". Obviously (X,≤) is a partially ordered set.
A sober space is a topological space such ...
CommunityBot
- 1
6
votes
1
answer
301
views
Irreducible of finite Krull dimension implies quasi-compact?
Let $X$ be the underlying space of a scheme.
If $X$ is irreducible of finite Krull dimension, is it necessarily
quasi-compact?
Is it necessarily Noetherian?
What if we assume not
only that Krull ...
- 16.2k
5
votes
1
answer
486
views
What is the connection between Frechet Lie groups and Lie algebras?
An ordinary Lie group has a differentiable manifold structure, i.e. it is locally isomorphic to a finite-dimensional Euclidean space. A Frechet Lie group, on the other hand, has a Frechet manifold ...
- 7,537
15
votes
1
answer
549
views
A continuum which is both Suslinean and non-Suslinean?
Continuum means compact connected metrizable with more than one point.
A continuum is Suslinean if every collection of pairwise disjoint subcontinua is countable.
There is an apparent contradiction ...
CommunityBot
- 1
10
votes
1
answer
385
views
Examples of Kreisel-Putnam topological spaces
Let us say that a topological space $X$ is a Kreisel-Putnam space when it satisfies the following property:
For all open sets $V_1, V_2$ and regular open set $W$ of $X$, if a point $x\in X$ has a ...
- 30.1k
9
votes
1
answer
384
views
The (co)tangent sheaf of a topological space
Let $X$ be a topological space (assume additional assumptions if needed) and denote by $\mathcal O _X$ its sheaf of $\Bbbk$-valued continuous functions where $\Bbbk$ is $\mathbb{R}$ or $\mathbb{C}$ ...
- 54.4k
4
votes
0
answers
137
views
Is there any quasi-compact space which is not a quotient of any compact Hausdorff space?
Is there any quasi-compact (= compact, possibly non-Hausdorff) space which is not a quotient of any compact Hausdorff space?
I strongly suspect the answer is yes, yet I couldn't come up with an ...
- 60.2k
2
votes
2
answers
175
views
Piecewise-metrizability problems from Willard's Topology
Maybe someone familiar with Willard's textbook can help me out. Problem section 23G on pg. 174 is titled piecewise metrizability. The first problem is:
If a Tychonoff space $X$ is the union of ...
CommunityBot
- 1
87
votes
19
answers
19k
views
Injectivity implies surjectivity
In some circumstances, an injective (one-to-one) map is automatically surjective (onto). For example,
Set theory
An injective map between two finite sets with the same cardinality is surjective.
...
CommunityBot
- 1
0
votes
1
answer
132
views
Sufficient and necessary condition for the global uniqueness of fix-points
https://www.ams.org/journals/proc/1976-060-01/S0002-9939-1976-0423137-6/S0002-9939-1976-0423137-6.pdf
This paper gives a sufficient condition for the uniqueness of SCHAUDER fix point. I wonder if ...
- 53.6k
4
votes
1
answer
236
views
Alexandrov's mapping lemma
I'm looking for a complete proof of Alexandrov's mapping lemma. I'd also like to have the intuition for it explained if that's at all possible. Or alternatively any pointers in the right direction ...
- 95.6k
8
votes
2
answers
545
views
A reference to a well-known characterization of scattered compact spaces
It is well-known that a compact Hausdorff $X$ space is scattered if and only if admits no continuous maps onto the unit interval $[0,1]$.
Surprisingly, but I cannot find a good reference to this well-...
- 40.9k
3
votes
3
answers
349
views
Diffeomorphism with prescribed behaviour
If $\gamma$ and $\eta$ are two smooth curves in a smooth manifold $M$, is it possible to find a diffeomorphism of $M$ such that $f \circ \gamma = \eta$? What if one removes the assumption of ...
- 312
7
votes
2
answers
264
views
The union of all coreflective Cartesian closed subcategories of $\mathbf{Top}$
Very often, in topology, one restricts to a coreflective Cartesian closed subcategory of $\mathbf{Top}$ in order to freely use exponential laws for mapping spaces, which imply things like "the ...
- 6,661
3
votes
1
answer
243
views
Category of continuous self maps
Is there any way to reconstruct a topological space from the category of its continuous self maps (possibly under some assumptions)?
How can we tell whether a category is the category of continuous ...
- 2,140
2
votes
2
answers
750
views
On One point Lindeloffication of topological spaces
As you Know when we define a topological space to be the one point compactification of the topological space $X$, we look for a compact space $Y$ such that $X\subset Y$ and $X$ is dense in $Y$ and $|Y-...
- 27.8k
4
votes
1
answer
667
views
Homotopy groups of fiber products
Let $X, Y, B$ be three smooth manifolds, and $f : X\to B$, $g : Y\to B$ submersions.
Then $X\times_BY$ exists.
(1) If $X, Y, B$ have the homotopy type of a finite CW complex, does $X\times_BY$?
(2) ...
CommunityBot
- 1
4
votes
0
answers
423
views
Finite good covers on smooth manifolds
Let $M$ be a connected smooth manifold that is not necessarily compact but has the homotopy type of a finite CW complex.
Does $M$ admit a finite good cover? (i.e. a finite cover by contractible ...
- 27.3k
14
votes
0
answers
419
views
A meager subgroup of the real line, which cannot be covered by countably many closed subsets of measure zero?
Is there a ZFC-example of a subgroup $H$ of the real line $\mathbb R$ such $H$ is meager, has zero Lebesgue measure, but cannot be covered by countably many closed subsets of measure zero in $\mathbb ...
- 40.9k
1
vote
0
answers
288
views
Fully faithful functor from schemes to spaces
Is there a fully faithful functor from the category of schemes
to the category of topological spaces and continuous maps (or some other sufficiently topological objects, e.g. smooth manifolds and ...
CommunityBot
- 1
2
votes
1
answer
641
views
Hairy ball theorem for odd-dimensional spheres
Let $\mathbb S^n$ be the $n$-sphere: $$\mathbb S^n=\left\{x \in \mathbb R^{n+1}: \left\|x\right\|=1\right\}.$$The hairy ball theorem can be formulated as follows:
If $n$ is even and $f\,\colon\, \...
CommunityBot
- 1
2
votes
1
answer
73
views
Are compactifications of completely $T_{4}$ spaces completely $T_{4}$?
The title is the question.
Given a locally compact completely $T_{4}$ space $X$ (every subspace is $T_{4}$) and a (Hausdorff) compactification $\overline{X}$ of $X$, is $\overline{X}$ also completely ...
- 5,317
4
votes
1
answer
243
views
Does each $\omega$-narrow topological group have countable discrete cellularity?
A topological space $X$ is defined to have countable discrete cellularity if each discrete family of open subsets of $X$ is at most countable.
A family $\mathcal F$ of subsets of a topological space ...
- 40.9k
0
votes
0
answers
59
views
A retract algebraic subset of the plane which does not admit an algebraic retraction
What is an example of an algebraic (=Zariski closed) subset $C$ of $\mathbb{R}^2$ which is a topological retract of $\mathbb{R}^2$, but there is no algebraic retraction $P:\mathbb{R}^2 \to C$?
What ...
- 60.2k
3
votes
1
answer
255
views
If the finitely additive measure of an open set is approximable by clopen sets, is it approximable from within?
Let $F$ be a finite set equipped the discrete topology. Let $X = F \times F \times ...$ be the countably infinite product space equipped with the product topology. Let $\mathcal A$ be any field of ...
- 839
4
votes
0
answers
584
views
A simple proof of Jordan curve theorem [closed]
I need a short proff of the Jordan curve theorem please.
The one I have is 16 pages long and is for a little expo, so I need one a little shorter.
Thanks
- 312
0
votes
0
answers
57
views
Does the total space of a bundle satisfy the Tietze extension property when the fiber and base space do satisfy this property?
We say that a Topological space $Y$ satisfies the Tietze extension propery, TE property, if in the formulation of Tietze extension theorem "$\mathbb{R}$" can be replaced by $Y$.
Obvioysly the ...
- 312
1
vote
1
answer
164
views
$n$-product-periodic topological spaces
We call an topological space $(X,\tau)$ $n$-product-periodic for an integer $n\geq 3$ if $\prod_{i=1}^n X \cong X$ but for all integers $k$ with $2\leq k\leq n-1$ we have $\prod_{i=1}^k X \not\cong X$....
- 4,618
40
votes
2
answers
2k
views
Ultrafilters as a double dual
Given a set $X$, let $\beta X$ denote the set of ultrafilters. The following theorems are known:
$X$ canonically embeds into $\beta X$ (by taking principal ultrafilters);
If $X$ is finite, then there ...
- 42.1k
9
votes
1
answer
334
views
How much can complexities of bases of a "simple" space vary?
Given a countable subbase of a topology, we can consider its complexity in terms of the difficulty of determining whether one family of basic open sets covers another basic open set. My question is ...
- 19.1k
1
vote
2
answers
154
views
Requirement for connected sets [closed]
Let $E$ be a compact metric space. Suppose that closure of every open ball $B(a,r)$
is the closed ball $B'(a,r)$. Must every open ball in $E$ be connected?
I think it most probably is. But I don't ...
- 52.4k
3
votes
0
answers
90
views
Is there a T3½ category analogue of the density topology?
Motivation: I understand that various attempts have been made at defining a topology on $\mathbb{R}$ that is an analogue of the density topology ([1]) but for category (and meager sets) instead of ...
- 30.1k
14
votes
4
answers
1k
views
Localic locales? Towards very pointless spaces by iterated internalization.
One can think of locales as (generalizations of) topological spaces which don't necessary have (enough) points. Of course when one studies locales, one "actually" studies frames,
certain sorts of ...
- 27.8k
5
votes
0
answers
102
views
Universal and strong $Q$-sets
A subset $X\subset \mathbb R$ is called
$\bullet$ a $Q$-set if for every subset $A\subset X$ there exists a $\sigma$-compact set $C\subset\mathbb R$ such that $C\cap X=A$;
$\bullet$ a strong $Q$-set ...
- 40.9k
11
votes
1
answer
385
views
The Parovichenko cardinal, is it equal to $\max\{\aleph_2,\mathfrak p\}$?
Let us define the Parovichenko cardinal $\mathfrak{P}$ as the largest cardinal $\kappa$ such that each compact Hausdorff space $K$ of weight $w(K)<\kappa$ is the continuous image of the remainder $...
- 40.9k
9
votes
1
answer
278
views
Reference request: filter tends to filter along map
Recall that a filter on a set $X$ is a nonempty collection $\mathcal{F}$ of subsets of $X$ such that
(i) $U\subseteq V\subseteq X$ and $U\in\mathcal{F}$ implies $V\in\mathcal{F}$, and
(ii) $U,V\in\...
- 27.8k