Questions tagged [gn.general-topology]
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
4,432
questions
4
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What does the Grothendieck topos tell us about the homotopy type of a space?
Let $M_1$, $M_2$ be two closed connected topological manifolds. We can consider the small sites of open coverings of them, and the categories of sheaves on these sites.
what can we say about $M_1$ ...
7
votes
1
answer
234
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Comparison of several topologies for probability measures
Let $X$ be a compact metric space and denote $\mathcal M(X)$ the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\operatorname{supp} \mu$ for the support of $\mu$. As is well ...
6
votes
2
answers
396
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 ...
9
votes
1
answer
837
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A Besicovitch-type Covering Theorem
In the book The Geometry of Domains in Spaces by Krantz and Parks, the authors proved the weak $(1,1)$-type estimate of the maximal function $M_\mu f$, where $\mu$ is a Radon measure, using their ...
1
vote
2
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184
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Reference request: lower sets of a preorder form a lattice
Consider a set $S$ with a preorder $\preceq$ (a preorder is a reflexive and transitive relation). A lower set $A$ of $S$ is defined as a subset of $S$ such that for all $x \in S$ and $y \in A$, if $...
5
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0
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202
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Is unitary group paracompact?
In this paper Martin Schottenloher notices that the unitary group $U(H)$ of a separable Hilbert space $H$ is metrizable in the strong operator topology. As a corollary (see R.Engelking, 5.1.3), it is ...
14
votes
1
answer
634
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Quotient of Three Dimensional Torus by Permutation on Coordinates
The Mobius Strip can be realized as a quotient of $T = (S^1)^2$ via the identifications $(x,y) \sim (y,x)$.
I tried to generalized this concept to a higher dimension, and consider the quotient of $(...
1
vote
1
answer
85
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Hausdorff quotient collapsing and separating a prescribed collection of disjoint closed subsets
Let $X$ be a compact Hausdorff space (I don't mind assuming it's metrizable).
Let $A_i$ $i\in \mathbb{N}$ be a collection of disjoint closed subsets of $X$.
My question: Does there exist a Hausdorff ...
3
votes
2
answers
382
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The underlying space of an affine open dense subscheme
Let $X$ be a Noetherian scheme, $U\subset X$ be an affine open dense subscheme. Is the underlying space of $U$ necessarily homeomorphic to the underlying space of $X$?
18
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0
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2k
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Characterization of Fréchet-Urysohn spaces using sequential continuity at a point
A map $f \colon X \to Y$ is called sequentially continuous at the point $a$ if for every sequence $(x_n)$ such that $x_n\to a$, we also have $f(x_n)\to f(a)$.
$$x_n\to a \qquad \Rightarrow \qquad f(...
14
votes
5
answers
6k
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Regular spaces that are not completely regular
In the undergraduate toplogy course we were given examples of spaces that are $T_i$ but not $T_{i+1}$ for $i=0,\ldots,4$. However, no example of a space which is $T_3$ but not $T_{3.5}$ was given. ...
2
votes
1
answer
186
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Quasi-compactification of locally spectral spaces
Let $X$ be a locally spectral topological space (i.e. a space admitting an open cover by spectral spaces). Does there necessarily exist a quasi-compact locally spectral space $Y$ and an injective ...
5
votes
0
answers
200
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An easier example of complete lattice such that the Scott topology on it is not sober
Basic notions: $1$, A partially ordered set is a dcpo if each of its directed subsets has a supremum. (https://en.wikipedia.org/wiki/Complete_partial_order)\
$2$, A subset O of a dcpo P is called ...
0
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0
answers
319
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Grothendieck topology on a scheme equivalent to the circle
Suppose a class of morphisms of schemes is reasonable enough so that we can associate a small site to any scheme (just like small étale site). Is there a simple class of morphisms such that there is a ...
2
votes
0
answers
229
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Refining monotone-light factorizations
Let $f:X\to Y$ be a continuous map between topological spaces. Consider the quotient map $\pi:X\twoheadrightarrow X/E$ given by decomposing the fibers of $f$ to their connected components.
In Lemma 6....
5
votes
0
answers
111
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Is there a homogeneous compactum where non-empty $G_\delta$s have non-empty interior?
A space $X$ is called an almost $P$-space if $Int(G) \neq \emptyset$ for every non-empty $G_\delta$ subset $G \subset X$.
Every $P$-space (that is, a space where $G_\delta$s are open) is an almost $P$...
1
vote
0
answers
294
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Closed and discrete sets
Let $\kappa$ in an uncountable regular cardinal and $X$ be a space and $e(x)=\kappa$, where the ``extent'' $e(X)$ of $X$ is the supremum of the cardinalities of
closed discrete subsets of $X$. My ...
1
vote
1
answer
185
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Countable union of Menger spaces
A topological space $X$ has Menger's property $\textsf{S}_{\mbox{fin}}(\mathcal{O}, \mathcal{O})$ if, for each sequence of open covers, $\mathcal{U}_1, \mathcal{U}_2, \cdots $, we can select finite ...
6
votes
6
answers
485
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If $(X,\tau)$ has more than $1$ point and is $T_2$ and connected, do we have $|X| =|\tau|$?
If $(X,\tau)$ has more than $1$ point and is $T_2$ and connected, do we necessarily have $|X| =|\tau|$?
4
votes
0
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103
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Borel selections of usco maps on metrizable compacta
The problem posed below is motivated by this problem of Chris Heunen and in fact is its reformulation in the language of usco maps. Let us recal that an usco map is an upper semicontinuous compact-...
6
votes
2
answers
192
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A non-Borel union of unit half-open squares
On the complex plane $\mathbb C$ consider the half-open square $$\square=\{z\in\mathbb C:0\le\Re(z)<1,\;0\le\Im(z)<1\}.$$
Observe that for every $z\in \mathbb C$ and $p\in\{0,1,2,3\}$ the set $...
7
votes
2
answers
484
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Do continuous maps factor through continuous surjections via Borel maps?
Let $f \colon X \twoheadrightarrow Y$ be a continuous surjection between compact Hausdorff spaces, and $g \colon \mathbb{R} \to Y$ a continuous function. Can you always find a Borel-measurable ...
11
votes
1
answer
771
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The (fiber of the) cofiber of the fiber of a map of spaces
Consider a fiber sequence of spaces
$$F \overset{i}{\to} E \to B$$
The cofiber $C(i)$ of the inclusion of the fiber comes with a canonical map $C(i) \to B$. Its possible to show (using some point ...
4
votes
0
answers
78
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Is there an $L$-space whose square is selectively $d$-separable?
An $L$-space is a hereditarily Lindelof regular space which is not separable.
A space is $d$-separable if it contains a dense set which is the countable union of discrete sets.
An $L$-space can't ...
4
votes
0
answers
81
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source of argument about relative primeness of simple closed curves on tori
I have know this argument for decades. I have no idea of its source. If anyone knows (not guesses) its origin, then I would be very appreciative. My guesses are among Ralph Fox, JHC Whitehead, RH ...
0
votes
0
answers
146
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A scheme whose underlying space is the product of the underlying spaces of schemes
We know that the product of two spectral topological spaces is spectral.
If $X$ is the underlying space of the scheme $\mathrm{Spec}\,\mathbb{Z}[x]$, what is a simple example of an affine scheme ...
2
votes
0
answers
125
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Homeomorphic extension to totally disconnected sets
Dear Mathoverflow Community,
I am looking for a reference for the following topological fact:
Fact
Let $E$ and $F$ be two totally disconnected compact subsets of the plane (can assume perfect if ...
4
votes
2
answers
1k
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topological group that is connected and locally connected but not path-connected
Is there a ($\mathrm{T}_0$) topological group that is connected and locally connected but is not path-connected?
This is a cross-post from MSE, since my question there was posted over three weeks ago ...
16
votes
0
answers
358
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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 ...
9
votes
1
answer
576
views
Is there a linearly Lindelöf non-Lindelöf $P$-space?
A completely regular topological space $(X,\tau)$ is a $P$-space, if every $G_\delta$-subset of $X$ is open (i.e $\tau$ is closed under countable intersections).
A topological space $X$ is linearly ...
5
votes
0
answers
227
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 ...
1
vote
0
answers
44
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Decomposition with given closures [closed]
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\,.$$
Does it ...
4
votes
1
answer
220
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K-analytic spaces whose any compact subset is countable
A regular topological space $X$ is called
$\bullet$ analytic if $X$ is a continuous image of a Polish space;
$\bullet$ $K$-analytic if $X$ is the image of a Polish space $P$ under an upper ...
3
votes
0
answers
132
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Is there a normal space with a $G_\delta$ diagonal which is not submetrizable?
A space has a $G_\delta$-diagonal if its diagonal can be written as the intersection of countably many open subsets of the square. A space is submetrizable if it has a weaker metrizable topology. ...
9
votes
0
answers
216
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Point-free topology, but with $\sigma$-algebras instead of spaces
I have a question about $\sigma$-algebras in relation to point-free topology. The question was inspired by a comment on a similar question I had:
If abstract $\sigma$-algebras (i.e. certain boolean ...
6
votes
0
answers
141
views
Countably compact non-compact perfect spaces
Recall that a space is countably compact if every infinite set has an accumulation point. A space is perfect if every closed set is a countable intersection of open sets. One of the classical ...
9
votes
4
answers
492
views
Infimum of a finite number of distances in the plane
Suppose one has a finite number of distances $d_1,\ldots,d_k$ on the Euclidean plane all of which metricize the usual Euclidean topology.
Define for each pair of points $x$ and $y$ in the plane
$$d(x,...
2
votes
1
answer
899
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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 ...
1
vote
0
answers
65
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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 $(...
6
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0
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352
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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 ...
2
votes
1
answer
101
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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 ...
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?
2
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0
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208
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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 $n$ ...
4
votes
0
answers
101
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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. ...
3
votes
1
answer
139
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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 ...
2
votes
1
answer
109
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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 $|O_x| < \...
8
votes
2
answers
911
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When does Scott topology generated by specialization order induced by a sober space (X,$\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 ...
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 ...
5
votes
1
answer
483
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 ...
15
votes
1
answer
549
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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 ...