Questions tagged [gn.general-topology]
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
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3 views
Aspect ratio of random walks from the gyration tensor
In the context of random walks or polymer chains, a useful quantity for capturing and characterizing the shape of the walk or the conformation of the polymer in space is the gyration tensor $T$, which ...
0
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0answers
31 views
Connected components of open dense subspace
Let $X$ be a sober topological space. Let $U\subset X$ be a dense open subset. Can the number of connected components of $U$ (in induced topology) exceed that of $X$?
5
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0answers
109 views
What is the smallest number of hyperplanes covering $\ell_2$?
For a Banach space $X\ne \{0\}$, let $\mathrm{cov}_H(X)$ be the smallest number of hyperplanes covering $X$.
By a hyperplane in a Banach space I understand any closed affine subspace of codimension ...
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0answers
29 views
Decomposition with given closures [on hold]
Let $(X,\mathcal T)$ be a topological space. About the subsets $A,B,C$ of $X$ it is known that $$\mathrm {cl} (C)= A \cup B\,, \quad \mathrm {cl} (A) = A\,, \quad \mathrm {cl} (B) = B\,.$$
Does it ...
8
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0answers
150 views
On projectively countable sets in the Hilbert cube
A subset $A$ of a topological space $X$ is called projectively countable if for any continuous map $f:X\to\mathbb R$ the image $f(A)$ is countable.
It is easy to see that each projectively countable ...
3
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0answers
64 views
A slightly canonical way to associate a scheme to a Noetherian spectral space
Let $C$ be the category whose objects are Noetherian spectral topological spaces and whose morphisms are homeomorphisms. Let $\mathrm{AffSch}$ be the category of Noetherian affine schemes (morphisms ...
3
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0answers
38 views
Topological properties of Noetherian affine schemes that do not hold for general Noetherian spectral spaces
I used to think that the only reason why an affine scheme with a Noetherian space can fail to be Noetherian is nilpotents. It turns out that this is not true.
This leads me to the following question:...
6
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0answers
92 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 ...
5
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0answers
72 views
Fractal plane continuum with order $\omega$?
Continuum means compact and connected.
The order $ord(x)$ of a point $x$ in a continuum $X$ is defined to be the least ordinal $\alpha$ such that $X$ has a neighborhood base of open sets at $x$ with ...
6
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0answers
104 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 ...
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1answer
41 views
How to determine the family of bounded functions from an infinite Fort space to $[0,1]$?
Definition: Let $X$ be a topological space and $b\in X$. We call $X$ a Fort space (with particular point $b$), when $X$ has topology $\{A\subseteq X: b \not\in A \; \text{or} \; X\setminus A\; \text{...
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51 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 $(...
5
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1answer
109 views
Do continuous maps factor through continuous surjections via Borel maps?
Let $f \colon X \twoheadrightarrow Y$ be a continuous surjection between compact Hausdorff spaces, and $g \colon \mathbb{R} \to Y$ a continuous function. Can you always find a Borel-measurable ...
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0answers
49 views
Version of Knaster-Tarski that gives unique fixed point
Can someone give a reference (if it exists) to a version of Knaster-Tarski that gives a unique fixed point?
6
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0answers
115 views
Do codimension 1 subsets of a scheme cover it?
Let $X$ be an irreducible scheme. A point $p\in X$ is said to have codimension $n\in\mathbb{Z}_{\geq 0}\cup \{\infty\}$ if $\overline{\{p\}}$ has codimension $n$. Is it true that any point of positive ...
2
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1answer
50 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 ...
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59 views
Topological spaces - Meager in itself [closed]
someone know examples of topological spaces of first category in themselves (meager in itself), that are not countable? Note that the Cantor set is nowhwere dense in $\mathbb{R}$ but it is not meager ...
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42 views
Homeomorphisms coinciding on closed irreducible subsets
Let $f_1$, $f_2:X\rightarrow Y$ be two homeomorphisms of sober spaces. Assume that for any closed irreducible subset $Z\subset X$, we have $f_1(Z)=f_2(Z)$. In particular, $f_1$ and $f_2$ coincide on ...
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1answer
75 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?
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0answers
37 views
Closed points on scheme locally of finite Krull dimension
Let $X$ be an irreducible scheme that has a possibly infinite cover by open sets of finite Krull dimension. Does $X$ have a closed point?
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0answers
38 views
Finiteness of Krull dimension is local
Let $X$ be an irreducible scheme. Suppose $X$ has a cover by affine opens that have Krull dimension bounded by some integer. Does $X$ have finite Krull dimension?
4
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0answers
69 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. ...
5
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0answers
142 views
Complexity of the set of closed subsets of an analytic set
Let $X$ be a compact Polish space and $K(X)$ the hyperspace of closed subspaces of $X$ with the Vietoris/Hausdorff metric topology.
Question: If $A$ is an analytic subset of $X$, what is the ...
3
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1answer
96 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 ...
2
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1answer
64 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| < \...
1
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1answer
107 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 $n$ ...
6
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1answer
242 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 ...
5
votes
1answer
76 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 ...
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42 views
Convergence of slices in some topology on hyperspace of closed sets
Let $X$ and $Y$ be metrizable spaces, $Y$ compact, and $C\subseteq X\times Y$ closed.
For each $y\in Y$, let $C^y=\{x\in X:(x,y)\in C\}$, the $y$-slice of $C$. Since $Y$ is compact, the projection ...
4
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1answer
152 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 ...
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2answers
96 views
A P-Cauchy sequence is it, a Cauchy sequence? [closed]
During my research I came across this notion, that I named a sequence of P-Cauchy (P for pseudo) :
We say a sequence $(x_n)_n$ of a metric space is P-Cauchy if :
$\forall e>0, \exists N \in \...
15
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1answer
462 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 ...
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0answers
42 views
Trying to understand pointwise convergence in the symmetry group of X [migrated]
Apologies for a question that I'm sure is very silly, but I'm not an expert on symmetry groups and this example is confusing me.
So, suppose we define the following functions from $\mathbb{R}\...
4
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0answers
81 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 ...
7
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1answer
275 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 $\mathbb{C}$ ...
2
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2answers
110 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 ...
10
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1answer
228 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 ...
2
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1answer
209 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 ...
0
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1answer
44 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 ...
4
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1answer
150 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 ...
8
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2answers
424 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-...
3
votes
1answer
201 views
Is the ring of $p$-adic integers extremally disconnected?
We call a topological space $X$ extremally disconnected if the closures of its open sets remain open. Obviously, Hausdorff extremally disconnected spaces are totally disconnected in the sense that ...
3
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1answer
214 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 ...
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87 views
Two small uncountable cardinals related to Q-sets
A subset $A$ of the real line is called a Q-set if any subset of of $A$ is of type $F_\sigma$ in $A$.
Let $\mathfrak q_0$ be the smallest cardinality of a subset $X\subset\mathbb R$ which is not a Q-...
2
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1answer
197 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 $X\times_BY$?
...
4
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0answers
131 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 ...
6
votes
2answers
168 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 ...
2
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0answers
195 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 ...
2
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1answer
65 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 ...
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0answers
62 views
Can closure and complement generate 14 distinct operations on a topological space if no subset generates more than 6 distinct sets?
$\newcommand{\XT}{(X,\mathcal{T})}$From Definition 1.2 in Gardner and Jackson (GJ): The $K$‑number $K(\XT)$ of a topological space $\XT$ is the cardinality of the Kuratowski monoid of operators ...
