Questions tagged [gn.general-topology]
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
4,447
questions
11
votes
0
answers
213
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Shift invariant measurable selection theorem
Let $(X,\mathcal{F})$ be some measure space and endow $\mathbb{R}^\mathbb{Z}$ with the product topology and borel $\sigma$-field. Let $F$ be a point to set mapping $X^\mathbb{Z}\rightarrow \mathcal{P}(...
2
votes
0
answers
553
views
Valuation topology vs modified valuation topology
Let $K$ be a field with valuation $v:K\to G\cup\{\infty\}$ where $G$ is an ordered abelian group. In section 7.62 of the book "Foundations of analysis over surreal number fields." Vol. 141. Elsevier, ...
2
votes
1
answer
324
views
Separability of $L^1$ in $L^2$ topology
In the space $L^1(0,1)$ take the topology generated by the $L^2$-balls
$$B^2_r(f)=\{g\in L^1(0,1):\; \|f-g\|_2<r\}.$$
Is $L^1(0,1)$ separable in this topology?
10
votes
2
answers
442
views
Quotient of $S^3$ by Montgomery and Zippin's "wild involution"
In 1952, Bing showed the existence of a topological involution of $S^3$ with fixed point set the Alexander horned sphere, demonstrating that $S^3$ has finite-order homeomorphisms not conjugate to ...
12
votes
1
answer
1k
views
What is the structure associated to almost-everywhere convergence?
Let $M(X)$ be the vector space (actually it's an algebra) of all equivalent classes of measurable functions $X\to \mathbb{C}$ (where $X$ is a measured space) modulo equality almost-everywhere.
One ...
12
votes
4
answers
1k
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Elementary proof that knot complements are path-connected
The complement of any (topological) knot is path-connected. More precisely, if $K$ is a subset of $\mathbb{R}^3$ (or $S^3$) homeomorphic to $S^1$, then $\mathbb{R}^3\setminus K$ (or $S^3\setminus K$) ...
1
vote
0
answers
135
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Finding a metric on a topological space with prescribed isometry group
Let $X$ be a (sufficiently nice) topological space and let $\mathcal{F}$ be a group of homeomorphisms of $X$. Assume that $\mathcal{F}$ is also closed under point-wise convergence. I would like to ...
3
votes
1
answer
190
views
When is a real-valued function on a metric space a metric?
Let $X$ be a space, $\ f:X\times X\rightarrow\mathbb{R}^+\cup\{0\}$ be a map satisfying the first two axioms for a metric (so $f(x,y)=0$ exactly when $x=y$, and $f$ is symmetric).
Now, consider the ...
10
votes
1
answer
2k
views
When does the sheaf cohomology of a topological space vanish?
The question is in the title. A more precise formulation is:
Let $X$ be a topological space. When does $H^i(X,F) = 0$ for all $i > 0$ and all abelian sheaves $F$ on $X$?
The obvious example is a ...
13
votes
1
answer
1k
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A topology on $\Bbb R$ where the compact sets are precisely the countable sets
QUESTION.
In there a topology on $\Bbb R$ where the compact subsets are precisely the countable subsets?
I am trying to create a counterexample to a certain claim, and I found that what I need is a ...
6
votes
1
answer
140
views
Maximality and non-Hausdorffness
We say that a non-$T_2$ topology $\tau$ on a set $X$ is maximal non-$T_2$ if every topology $\tau'$ strictly containing $\tau$ is $T_2$.
Is every non-$T_2$ topology contained in a maximal non-$T_2$ ...
2
votes
1
answer
193
views
Explicit description of the closure of a given set
Let $C$ be the subset of $C_b(\mathbb{R})$ given by
$$C:=\{f\in C_b(\mathbb{R}):\ \exists f'\in C_b(\mathbb{R})\}$$
Now I want to take the closure of this set with respect to the supremum norm on $...
2
votes
1
answer
334
views
One-dimensional topological spaces
We know that all connected (not a singleton) subsets of $\mathbb{R}$ (with the usual topology) has no empty interior. This fact does not remains true for a general connected topological space with the ...
3
votes
1
answer
150
views
Topological spaces with Lebesgue covering dimension 1
We know that all connected subsets of $\mathbb{R}$( with the usual topology) has no empty interior. I would like to know if this fact remains true for a general
connected topological space with the ...
16
votes
3
answers
3k
views
Physical interpretations/meanings of the notion of a sheaf?
I fairly understand the fiber bundles, both the mathematical concept of fiber bundles and the physics use of fiber bundles. Because the fiber bundles are tightly connected to the gauge field theory in ...
2
votes
0
answers
77
views
Sheaf of R-modules and modules over compactly supported functions
I'm looking for a reference for the following result:
Let $X$ be a locally compact Hausdorff topological space. let $\mathcal{R}$ be the sheaf of continuous functions with values in $\mathbb{R}$ over ...
33
votes
4
answers
7k
views
Topology of function spaces?
Let $X,Y$ be finite-dimensional differentiable manifolds, and let's assume that they are connected. In fact, in applications I would like both $X$ and $Y$ to be riemannian manifolds.
Let $C^\infty(X,...
1
vote
1
answer
306
views
Convergent sequences in compact spaces
Problem. Assume that a compact space $X$ can be written as the union $X=K\cup D$ of a compact metrizable subspace $K$ and a discrete subspace $D$.
Does $D$ contain a non-trivial convergent sequence in ...
0
votes
1
answer
219
views
Does every compact countable space contain a non-trivial convergent sequence?
Problem. Does every compact countable space contain a non-trivial convergent sequence?
This question concerns non-Hausdorff compact spaces. An example of such space is any infinite set $X$ endowed ...
11
votes
2
answers
1k
views
How to show that something is not completely metrizable
I have a Polish space $X$ and a subset $A \subset X$.
I know that $A$ is completely metrizable (in its induced topology) if and only if $A$ is a $G_\delta$-set in $X$.
This means: If I want to show ...
3
votes
1
answer
234
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Characterisation of paracompact spaces by some sort of embeddability?
This question was inspired by this question.
Before I start, I don't really mean embedding in what follows. I'm tempted to use plongement, for an exotic touch, but well, that's just a rose by another ...
20
votes
2
answers
1k
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The Gelfand duality for pro-$C^*$-algebras
The Gelfand duality says that
$$X\to C(X)$$
is a contravariant equivalence between the category of compact Hausdorff spaces and continuous maps and the category of commutative unital $C^*$-algebras ...
2
votes
0
answers
109
views
Compare two topologies: Three 2-tori inside $S^3 \times S^1 \# S^2 \times S^2$ glued from two different diffeomorphisms
We like to ask for the comparison of two topologies of three 2-tori inside the same 4-manifolds glued from two different diffeomorphisms (see the end).
Given an embedded torus $T$ with trivial normal ...
3
votes
2
answers
178
views
A conjecture on antipodes and Jordan curves on the sphere
I got the following conjecture: Let $C$ be a Jordan curve on the sphere $S:=S^2$ and let $A$ and $B$ be the connected compnents of $S-C$. Then there is a pair of antipodes $a$ and $b$ such that $a\in ...
2
votes
1
answer
247
views
What lattices are isomorphic to $R^{N}$ for some $N$, equipped with the product order?
What lattices are isomorphic to $\mathbb{R}^{N}$ for some $N\in \mathbb{N}$, equipped with the canonical order?
Remark:
When I say $\mathbb{R}^N$, I don’t mean it to be a vector space. Instead, I ...
12
votes
0
answers
379
views
L-spaces without convergent sequences
An L-space is a regular hereditarily Lindelof space which is not hereditarily separable. Consistent examples of L-spaces are relatively easy to come by (for example, Suslin Lines), but the first ...
1
vote
0
answers
77
views
Random variables with values in binary operations or in topologies of a certain set $X$
I wonder if the following situations have already been considered by mathematicians :
Random variables with values in a set of binary operations endowed
with a certain topology (or just with a $\...
2
votes
1
answer
547
views
Is there a minimal, topologically mixing but not positively expansive dynamical system?
Is there a compact metric space $X$ and a function $f:X\to X$ such that the dynamical system $(X, f)$ has the following three properties?
minimal
topologically mixing (a map $f$ is topologically ...
6
votes
2
answers
647
views
Is there a topologically mixing and minimal homeomorphism on the circle (or on $\mathbb S^2$)?
The irrational rotation on the circle is both a homeomorphism and minimal but is not topologically mixing. The argument-doubling transformation on the circle is topologically mixing but is neither a ...
2
votes
1
answer
106
views
Is every topology the intersection of the $T_0$-topologies containing it?
This is a continuation of the question about Minimal $T_0$-spaces .
Let $X\neq \emptyset$ be a set and let $\text{Top}(X)$ denote the lattice of all topologies on $X$ and let $\tau\in\text{Top}(X)$.
...
5
votes
1
answer
319
views
Ramified covers of S^n
This question has been inspired by covering 3-torus post.
Is it true that any good (smooth, compact, oriented) $n$-manifold can be mapped to $S^n$ in such a way that the map is true covering away ...
5
votes
1
answer
474
views
Hausdorff dimension of boundaries of open sets diffeomorphic to $\mathbb{R}^n$
Let $B$ be a bounded open subset of $\mathbb{R}^n$ which is diffeomorphic to $\mathbb{R}^n$. (I am not sure how important the diffeomorphism is but this is the case I am interested in.) Let $C$ be its ...
23
votes
6
answers
2k
views
Is there a topological description of combinatorial Euler characteristic?
There are a collection of definitions of "combinatorial Euler characteristic", which is different from the "homotopy Euler characteristic". I will describe a few of them and give some references, and ...
1
vote
1
answer
172
views
Kelley & Namioka's definition of topology of uniform convergence on a subset
In Kelley & Namioka's Linear Topological Spaces, they begin section 8 on Function Spaces with a definition of the topology of uniform convergence. I've reproduced the begining of the first ...
4
votes
1
answer
131
views
Image spaces of $X^2$
Let $(X,\tau)$ be a topological space and let $$\Omega(X)=\{A\subseteq X: \text{there is surjective continuous }f:X\to A\}.$$
Can we express $\Omega(X\times X)$ in terms of $\Omega(X)$? (We assume $X\...
9
votes
0
answers
945
views
Topologies on compactly supported functions
Let M be a (non-compact) smooth manifold and consider the set $C^\infty_c(M)$ of smooth real-valued functions with compact support. We can give this function space several topologies. Here are four:
...
1
vote
2
answers
115
views
Are two pairs $(M\times M, M\times \{a\})$ and $(M\times M, D_{M})$ homeomorphic?
What is an example of a compact manifold $M$ without boundary which does not satisfy the following property:
For every $a\in M$, two pairs $(M\times M, M\times \{a\})$ and $(M\times M, D_{M})$...
19
votes
2
answers
3k
views
How bogus is the glitzy proof of Borsuk-Ulam?
Suppose $f: S^2 \rightarrow {\bf R}^2$ is continuous; let $A$ be the set of points $u \in S^2$ such that $f(u)-f(-u) \in {\bf R} \times \{0\}$ (where $-u$ denotes the antipode of $u$). Given $u,-u \in ...
0
votes
1
answer
568
views
Monotone convergence theorem for operators in the weak operator topology
For real numbers, we know that any monotonic bounded sequence converges to a finite limit. Does this generalize to sequences of operators?
More formally, I have a sequence of operators $\{A_n\}_{n=1}^...
6
votes
4
answers
2k
views
A simpler proof that compact sets have cardinality continuum?
Is there a simple reason why uncountable compact sets of real numbers have cardinality continuum?
I know that this is immediate from the Cantor-Bendixon Theorem, but I wonder whether this consequence ...
2
votes
1
answer
162
views
What is the most symmetric configuration of four 2-surfaces linked in $S^4$?
What are some of the most symmetric configurations of four 2-surfaces linked in the 4-dimensional sphere $S^4$?
To make a lower-dimensional analogy, recall that in 3-dimensional sphere $S^3$, we can ...
3
votes
0
answers
319
views
Cubical approximation theorem for cubical complexes
A version of the simplicial approximation theorem states that a continuous map between finite simplicial complexes is homotopic to a simplicial map after subdividing the domain.
I have found a claim ...
1
vote
1
answer
171
views
Interval topology on complete Boolean algebras
Let $(P,\leq)$ be a poset. The interval topology $\tau_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\...
8
votes
0
answers
121
views
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 ...
12
votes
1
answer
779
views
Restrictions of null/meager ideal
Let I denote the null (resp. meager) ideal on reals. Is it consistent that for any pair of non null (resp. meager) sets A and B, there is a null (resp. meager) preserving bijection between A and B? In ...
21
votes
2
answers
1k
views
An order type $\tau$ equal to its power $\tau^n, n>2$
(This is a re-post of my old unanswered question from Math.SE)
For purposes of this question, let's concern ourselves only with linear (but not necessarily well-founded) order types.
Recall that:
$...
21
votes
1
answer
1k
views
Which complete Boolean algebras arise as the algebras of projections of commutative von Neumann algebras?
Projections in an arbitrary commutative von Neumann algebra form a complete Boolean algebra.
Moreover, a morphism of commutative von Neumann algebras induces
a continuous morphism of the corresponding ...
8
votes
3
answers
588
views
Is there a non-metrizable topological space for which any countably compact subset is compact?
The title is the question : Is there a non-metrizable topological space for which any countably compact subset is compact ?
EDIT : non-metrizable and Hausdorff
3
votes
0
answers
82
views
Is the increasing union of disk bundles a disk bundle?
Setup: Let $B$ be a $C^r$ $n$-manifold ($r \geq 1$) and $M$ a closed $k$-dimensional $C^r$ submanifold of $B$. Assume there exists a smooth retraction $p:B \to M$ which is also a submersion, so that $...
8
votes
1
answer
771
views
When the boundary of any subset is compact?
Let $X$ be a Tychonoff space with no isolated points such that the boundary of any subset of $X$ is compact. Does it mean that $X$ is compact ? (If $X$ is a resolvable space then it is clearly compact....