Questions tagged [gn.general-topology]
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
1,155 questions with no upvoted or accepted answers
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H-spaces without rational homology
Does there exist a simply connected, non-contractible manifold $M$, which is an $H$-space,
and whose rational homology groups vanish in positive degrees?
My space $M$ is in fact homotopy equivalent ...
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Pseudocycle definition of open Gromov-Witten invariants
I decided my original question was unnecessarily long, and have edited to simply ask the desired question directly (the below is much more direct than my original post, though it may seem just as long!...
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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 ...
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Characterization of Unusual Topologies of $\mathbb R$
Following some argument over a question on math.SE, I began to wonder:
We all know that in the standard topology there are no continuous bijections from $[0,1]$ to $(0,1)$ (for example by arguments ...
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Maps with small fibers between manifolds of equal dimension
The following question is an attempt to revise this one into what I intended.
Important revisions are shown in bold.
Are there any known examples of a compact Riemannian manifold $M$ with (possibly ...
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A variation of necklace splitting
Our problem is the following:
Let $n$ and $k$ be integers.
We are given two (unclasped) necklaces, each with $n$ colored stones: a top necklace which has $k$ colors and a bottom necklace which has 2 ...
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Is $L^2(I,\mathbb Z)$ homeomorphic to the Hilbert space?
I am somehow puzzled by the subset $G:=L^2(I,\mathbb Z)$ of $H:=L^2(I,\mathbb R)$ of all integer valued functions on $I=[0,1]$ (in fact I mentioned as an example in this old MO question).
Some simple ...
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The pro-discrete space of quasicomponents of a topological space
Let $X$ be a topological space.
Consider the functor $P^X : \textbf{Set} \to \textbf{Set}$ that sends each set $Y$ to the set of continuous maps $X \to Y$.
It is not hard to check that $P^X : \textbf{...
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First order formula describing connected components
I ask this question here after no answer came up in the original MathSE question.
Let $\mathcal{L}$ be the language $\{+,-,\cdot,0,1,P\}$ where $P$ is some $n$-ary relation symbol. Is there a formula $...
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A modified version of the converse to the Sard's Theorem
When I learned Sard's Theorem in differential topology by myself, I was thinking whether it would be possible to prove a converse version of the theorem. That is to say, can we somehow show that each (...
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On "linearly independent" metric spaces
Urysohn's universal metric space $\Bbb U$ satisfies the following surprising property:
Whenever $i\colon\Bbb U\to E$ is an isometric embedding into a normed vector space such that $0\not\in i(\Bbb U)$...
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When is a constructible set locally closed?
Let $X$ be a topological space (or more specifically, $\mathbb{C}^N$ endowed with the Zariski topology), and let $S \subseteq X$ be a constructible set, i.e. $S=\cup_{i=1}^n C_i \cap U_i$, where the $...
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Continuous functions on a compact $T_1$ space
Let $X$ be a compact $T_1$ topological space consisting of more than one point, and suppose that $X$ is locally compact (i.e. every point has a local base of compact neighbourhoods), second countable, ...
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Topology and infinite number of primes
One of the strange proofs (among the other beautiful proof) in the book "Proofs from the book" is the fifth one, which uses a special topology on the set of integer numbers, to prove there are ...
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Topological applications of $\mathfrak{p}=\mathfrak{t}$
I'm working on the Malliaris-Shelah's recent result of $\mathfrak{p}=\mathfrak{t}$, but I'm more interested in what possible topological applications can derivate from this equality.
Searching in ...
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Universally meager spaces and large cardinals
Definition: (Todorcevic) A subset $A$ of a topological space $X$ is called universally meager if for every Baire space $Y$ and every continuous $f : Y \to X$ which is nowhere constant (not constant on ...
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Connected component optimization
For an open set $A\subset[0,1]^d$, denote the connected components of $A$ by $cc(A)$. Given a smooth symmetric function $f\colon[-1,1]^d\to\mathbb R$ with $f(0)>0$, I am interested in the ...
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"Generic properties" of open neighborhood boundaries in compact metric spaces
Suppose we have a compact metric space $X$ with some designated point $a$ and closed set $B$ such that $a\notin B$. Let $A_0=\{a\}$ and $B_0 = B$. We'll play a game where on each player's turn they ...
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Is the property of being a connected component local?
More precisely, my question is as follows: Let $X$ be a qcqs scheme, $Z \subset X$ a closed subscheme and assume that there exists an open affine subscheme $U \subset X$ containing $Z$ such that $Z$ ...
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How should I try to imagine exotic smoothness in R4?
I am trying to wrap my mind around the concept of exotic smoothness in (and only in) $\mathbb{R}^4$.
I have some mathematical literature, but can anyone point to a semi-intuitive, semi-visual example?...
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When can we force two frames to be homeomorphic?
Recall that if $M,N$ are two structures of the same type, then
$M$ is $\mathcal{L}_{\infty,\omega}$ elementarily equivalent to $N$ precisely when $M$ and $N$ are isomorphic in some forcing extension. ...
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Local vs global homogeneity of topological spaces
I am interested in the relation between global and local homogeneity of topological spaces. On one extreme we have rigid spaces, i.e., topological spaces with trivial homeomorphism group.
Question. ...
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Has the Roelcke completion of a topological group any reasonable algebraic structure?
It is well-known that each topological group $G$ carries (at least) four natural uniformities:
the left uniformity $\mathcal L$, generated by the base $\{\{(x,y)\in G\times G:y\in xU\}:U\in\mathcal ...
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Loop space functor and sequential colimits of inclusions
The question is about a fact that is mentioned as "evident" everywhere in the literature, so my guess is that some small detail is passing over my head. Here it is:
Let $X_0\hookrightarrow X_1 \...
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When is the sigma-algebra generated by closed convex sets the same as the Borel sigma-algebra
For which topological vector spaces $E$ do we have the equality between the sigma algebra generated by the closed convex subsets of $E$ and the Borel sigma algebra of $E$ ?
More precisely, do we have ...
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570
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example of an n-transitive but not infinitely transitive group action on a space
Definition. An action of a group $G$ on a set $X$ is strongly $n$-transitive if $G$ acts transitively on $n$-tuples of distinct elements in $X$ (via the diagonal action), and is $n$-transitive if $G$ ...
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Is there a normal non-collectionwise Hausdorff manifold?
In a 1990 paper*, M.E. Rudin writes (p.137),
So far as is known, normal manifolds may have to be collectionwise Hausdorff [cwH].
Since it holds whenever $V=L$, I understand that at that time, no ...
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Is each Lindelof closed $\bar G_\delta$-set of a Tychonoff space functionally closed?
A subset $F$ if a topological space $X$ is called functionally closed if $F=f^{-1}(0)$ for some continuous map $f:X\to[0,1]$.
It is clear that each functionally closed set $F$ in $X$ is a closed $G_\...
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Topology of family of complex varieties
It seems to be an oft-cited fact (which comes up, for instance, in describing vanishing/nearby cycles) that:
For a proper flat map $f \colon X \rightarrow \Delta$, where
$X$ is a complex algebraic ...
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Locales satisfying DC?
Is there a nice (topological) characterization of the locales such that the axiom of dependant choices holds in the internal logic of the topos of sheaves ? I would also be interested in the case of ...
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The Klee Trick for subsets of $\mathbb{R}^3$
Update: The lead paragraph has been changed to reflect the solution to a related question.
I asked the question Is dimension given by the Klee trick ever sharp?
and it has been answered in the ...
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Intersections of open sets and $\alpha$-favorable spaces
I would like to ask about some classes of topological spaces whether they have been studied, what are they called (if they have a name) and whether they have some interesting properties. For the ...
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Convex hulls of compact sets
Let $A$ be a compact set in a separable Hilbert space $H$, and let $\bar A$ denote its convex hull. Is $\bar A$ compact?
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Spaces that never separate the Hilbert cube
I am interested in topological spaces such that whenever the space embeds into the Hilbert cube, the image of the embedding has a path-connected complement.
Any finite dimensional space has this ...
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Is there in ZFC a topological space which is normal, ccc, countably compact, first countable and non-compact?
I am looking for a space as in the title and since many very similar spaces do exist in the literature, I wonder whether someone has a reference (different from the ones I cite below) or just some ...
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Construct a topologically $\infty$-dimensional separable metric space.
But don't assume knowledge of any topological dimension theory. Here is a specific approach (an open problem):
Does there exist a separable metric space $X$ such that the following two conditions ...
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Is the product of a discretely Lindelöf space with [0,1] discretely Lindelöf ?
A space $X$ is discretely Lindelöf iff given any discrete subset $D$ of $X$, its closure in $X$ is Lindelöf. Such spaces were introduced by Arkhangel'skii about 15 years ago (if I am not mistaken) ...
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Is there a generalization of Brouwer's fixed point theorem?
In essence, this is the same problem as in
“The generalization of Brouwer's fixed point theorem?”.
But now I am determined to be careful. The main question is
the following:
Is there any ...
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In a locally contractible space can we find local bases of contractible sets whose closures are locally contractible?
In a locally contractible topological space $X$ is it possible at each point $x$ to find a local basis of contractible sets $U_i\ni x$ such that the closure of each set $\overline{U_i} \subset X$ is ...
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An open set which is not the union of a closed set and a countable set
The following fact is probably a known result:
Fact. Let $X$ be an uncountable Polish space. Then there exists an open subset of $X$ which is not the union of a closed set and a countable set.
Proof:...
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Is there a Hausdorff space with a $\sigma$-locally finite basis but no $\sigma$-discrete basis?
In short, the question is in the title: is there a Hausdorff space with a $\sigma$-locally finite basis but no $\sigma$-discrete basis?
A bit of context:
Given a topological space $X$, a family $\...
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Generalizing uniform structures as Grothendieck topologies
Recently, I was reading a classical book "Sheaves in Geometry and Logic" by S. MacLane and I. Moerdijk, and then it stroke me that, that the definition of Grothendieck Topology bears some ...
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The space of analytic associative operations
This question is a follow-up to this old one of mine.
Let $\mathcal{A}$ be the set of functions $\star:\mathbb{R}^2\rightarrow\mathbb{R}$ which are associative and $C^\omega$ (real analytic entire) in ...
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The smallest cardinality of a cover of a group by algebraic sets
$\DeclareMathOperator\cov{cov}$A subset $A$ of a semigroup $X$ is called algebraic if $$A=\{x\in X: a_0xa_1x...xa_n=b\}$$ for some $b\in X$ and $a_0,a_1,...,a_n \in X^1=X\cup \{1\}$. The smallest ...
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Algebraizing topology and analysis via condensed mathematics
I asked this question on Mathematics Stackexchange, but one of the users suggested that I ask this question at MathOverflow.
I've just come across a Twitter thread by Laurent Fargues explaining a work ...
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A weak analogue of smooth manifolds (reformulated)
It is widely known that $C^1$ manifolds are topological spaces locally homeomorphic to Euclidean spaces and possessing $C^1$ chart-converters. They have a tangent space at every point, regarding as ...
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A locally compact, complete metric space in which the closure of open balls coincide with the closed ball is Heine-Borel
I saw the following result stated without a proof in a paper about the isometry group of metric measure spaces:
Let $X$ be a locally compact, complete metric space such that for all $x \in X$ and $R &...
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Is the derivative the unique operation on points in the plane that preserves convexity?
Let $C(n)$ be the space of multisets of size $n$ of points in the Euclidean plane, topologised appropriately, and consider a surjective continuous map: $$D:C(n)\rightarrow C(n-1)$$
Such that the ...
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Do algebraic completion/topological completion of fields always terminate? If so, are they unique?
Take the field $\mathbb{Q}$, If we complete it topologically with respect to the Euclidean norm, we get $\mathbb{R}$, then if we complete it algebraically, we get $\mathbb{C}$.
On the other hand, the ...
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Bounded open sets with same boundaries
Let $U_1$, $U_2$ two bounded open subsets of the euclidean plane.
and denote by $\partial U_1$ and $\partial U_2$ their topological boundaries.
Does $\partial U_1 = \partial U_2$ implies $U_1 = U_2$?
...
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