All Questions
5,184 questions
30
votes
5
answers
4k
views
The role of ANR in modern topology
Absolute neighborhood retracts (ANRs) are topological spaces $X$ which, whenever $i\colon X\to Y$ is an embedding into a normal topological space $Y$, there exists a neighborhood $U$ of $i(X)$ in $Y$ ...
- 18.7k
10
votes
0
answers
265
views
Let $X$ be a finite set of $n$ ($>1$) elements and $\tau$ be a topology on $X$ having exactly $m$ elements. Can we give any description of $m$?
Let $X$ be a finite set of $n$ ($>1$) elements and $\tau$ be a topology on $X$ having exactly $m$ elements.
Can we give any description of $m$ as it relates to $n$?
Obviously $2\le m\le 2^n$ and ...
- 53.3k
48
votes
6
answers
4k
views
Why the "W" in CGWH (compactly generated weakly Hausdorff spaces)?
In his 1967 paper A convenient category of topological spaces,
Norman Steenrod introduced the category CGH of compactly generated Hausdorff spaces
as a good replacement of the category Top topological ...
- 351
2
votes
1
answer
165
views
Hereditarily locally connected spaces
A topological space is locally connected if every point has a neighborhood basis of connected open subsets.
A property of topological spaces is termed hereditary, subspace-hereditary, if every subset ...
- 1,392
4
votes
0
answers
179
views
What's the unspoken history of compactly generated topological spaces?
Usually, the alleged motivation for the definition of compactly generated topological spaces is Cartesian closedness, which fails for general spaces. Of course, from a contemporary perspective, this ...
- 321
3
votes
2
answers
203
views
Recovering a set from its projections in varying coordinate systems - a projection hull?
Let me describe the simplest non-trivial case of what I have in mind. Let $V$ be a 2-dimensional $\mathbb{R}$-vector space and fix an isomorphism $V \cong \mathbb{R}^2$, where $\mathbb{R}^2$ is ...
- 55.9k
1
vote
0
answers
192
views
Simple left earthquakes are dense
i´ve been studying an article from W. P. Thurston about hyperbolic geometry, there, he defines something called left earthquake, whose definition is as follows:
Definition. If $\lambda$ is a geodesic ...
6
votes
1
answer
244
views
The continuity of certain maps on compact Hausdorff spaces
Let $f:M\to Y$ be a continuous proper bijective map from a metrizable space $M$ onto a $T_1$-space $Y$. The properness of $f$ means that for every compact subspace $K\subseteq Y$ the preimage $f^{-1}[...
- 41.9k
3
votes
0
answers
69
views
Is every weakly $1$-dimensional space embeddable in the plane?
A $1$-dimensional (separable metric) space $X$ is weakly $1$-dimensional if $$\Lambda(X)=\{x\in X:X\text{ is 1-dimensional at }x\}$$
is zero-dimensional (i.e. the space $\Lambda(X)$ has a basis of ...
- 3,317
17
votes
1
answer
569
views
Does a completely metrizable space admit a compatible metric where all intersections of nested closed balls are non-empty?
(cross-posted from this math.SE question)
It is well-known that given a metric space $(X,d)$, the metric is complete if and only if every intersection of nested (i.e. decreasing with respect to ...
- 41.9k
1
vote
0
answers
103
views
"Classifying" causally closed sets in Minkowski space
Let $M = \mathbb R^{D+1}$ be Minkowski space. Recall that the causal complement of a set $A \subseteq M$ is the set $A^\perp \subseteq M$ where $p \in A^\perp$ there is no timelike path between $p$ ...
- 64k
78
votes
12
answers
12k
views
Why aren't representations of monoids studied so much?
It seems to me like every book on representation theory leaps into groups right away, even though the underlying ideas, such as representations, convolution algebras, etc. don't really make explicit ...
- 39
6
votes
1
answer
259
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are endomorphisms "small" compared to the full transformations?
$\DeclareMathOperator\End{End}$Let $T_n$ be the full transformation semigroup/monoid of $[n]=\{1,\dots,n\}$. Let $\End(T_n)$ be the set of [endomorphisms][1] of $T_n$. Then, $\# T_n=n^n$ and
$$\# \End(...
- 63.9k
47
votes
4
answers
4k
views
Which topological spaces admit a nonstandard metric?
My question is about the concept of nonstandard metric space that would arise from a use of the nonstandard reals R* in place of the usual R-valued metric.
That is, let us define that a topological ...
- 11.4k
1
vote
1
answer
177
views
B-topological ring that is not semi-topological?
A topological space $A$ that is also a ring with operations ‘$+$’ and ‘$.$’, is a semi-topological ring if the mappings
\begin{gather*}
a_1: A\times A \to A\text{ such that }(x,y)\to x+y, \\
a_2: A \...
- 12.9k
3
votes
0
answers
107
views
Does the pseudo-arc contain Erdős space?
The pseudo-arc is the unique hereditarily indecomposable chainable continuum.
The Lelek fan is the unique compact, connected subset of the Cantor fan (the cone over the Cantor set) with a dense ...
- 3,317
7
votes
3
answers
1k
views
Connected space being not locally connected at each point
Say that a topological space $X$ is locally connected at some point $x$, if it has a local base at that point consisting of connected open sets.
Also $X$ is locally connected if it is locally ...
- 351
7
votes
1
answer
262
views
Can you remove a zero dimensional subspace from a cube and obtain a planar space?
The question, which came up in a conversation with my advisor Ola Kwiatkowska, is pretty much in the title:
Let $Z\subseteq[0,1]^3$ be zero-dimensional. Is it possible for $[0,1]^3\setminus Z$ to be ...
- 770
4
votes
0
answers
114
views
Find at least one square-boxed subcontinuum
Recall that a plane continuum is a closed, bounded,
connected subset of the plane.
It is non-degenerate if it contains at least two points.
(We may sometimes just say "continuum" even if
we ...
- 1,375
22
votes
1
answer
712
views
Are $\beta \mathbb{Q}$ and $\beta(\beta\mathbb{Q}\setminus\mathbb{Q})$ homeomorphic?
The canonical inclusion $\beta\mathbb{Q}\setminus \mathbb{Q} \hookrightarrow \beta\mathbb{Q}$ is not the Stone-Čech compactification of $\beta\mathbb{Q}\setminus \mathbb{Q}$. Even so, this doesn't ...
- 12.9k
2
votes
1
answer
360
views
Equivalence of the definitions of exactness and mixing
Let $f:X \to X$ be a continuous map, where $X$ is a compact metric space. We say that $f$ is (locally) expanding if there are constants $\lambda >1$ and $\delta_0 > 0$ such that, for all $x, y\...
- 136
1
vote
1
answer
161
views
Is there a two-dimensional unimodal function with fractal level sets
Is there an open simply connected $U\subset\mathbb{R}^2$ and a continuous non-constant function $f: U\to \mathbb{R}$,
such that for all $c\in \mathbb{R}$ both sets
$$ f_{<c}~=~ f^{-1}\left( (-\...
- 1,676
0
votes
0
answers
81
views
Is it possible to continuously embed $C^\infty(\mathbb{T}^n)$ as a vector space into $\mathcal{D}(\mathbb{R}^n)$ by some "inverse" of periodization?
Let $\mathbb{T}^n$ be the $n-$dimensional torus and $C^\infty(\mathbb{T}^n)$ be the Frechet space of smooth periodic functions on $\mathbb{R}^n$.
According to p.298 of Folland "Real Analysis"...
- 3,477
4
votes
1
answer
146
views
When is semigroup algebra local?
Let $G$ be a finite semigroup (or monoid if that helps) and $K$ a field.
Question: When is the semigroup algebra $KG$ local?
Here local means that there is a unique maximal right (or left) ideal.
...
- 38.6k
11
votes
1
answer
355
views
Name for topological spaces where "every point has a local base wellordered by reverse inclusion"?
There are many properties regarding local bases of a topological space, like first countable if every point has a countable local base.
Is there a similar name for a space where "every point has a ...
- 103
5
votes
1
answer
381
views
Sufficient criteria for $X \subset \mathcal{H}$ to be a Lipschitz (or unif. cont.) retract of $\mathcal{H}$
I am interested in sufficient criteria which ensure that a subset $X$ of a Hilbert space $\mathcal{H}$ is a Lipschitz (or at least uniformly continuous) retract of $\mathcal{H}$.
Under which ...
CommunityBot
- 1
6
votes
1
answer
229
views
Pixley and Roy article request
I'm trying to get a digital copy of the article "C. Pixley and P. Roy, Uncompletable Moore spaces,
Proc. Auburn Univ. Conf. (Auburn, Alabama, 1969), 75-85." but I have not been successful. ...
- 1,392
3
votes
1
answer
147
views
What exactly is the topology on $O_M$ that makes the convolution map $S \times S' \to O_M$ hypocontinuous?
Let $O_M(\mathbb{R}^n):= \mathcal{S}'(\mathbb{R}^n) \cap C^\infty(\mathbb{R}^n)$ be the space of slowly increasing smooth functions on $\mathbb{R}^n$.
Following p.294 proposition 9.10 of the "...
-1
votes
3
answers
523
views
Metric properties for $d:X\times X\times\dotsb X\rightarrow\mathbb R$ [closed]
Let us define $d:X^n\rightarrow\mathbb R$. How can we define metric properties such as symmetry, triangle inequality equivalent property etc for such a function?
- 12.9k
2
votes
0
answers
156
views
Do Grothendieck topoi with enough points satisfy the fan theorem internally?
Fourman and Hylland proved in the 80s that all spatial topoi satisfy the full fan theorem internally, while there are examples of localic topoi that do not satisfy it.
This leads one to conjecture a ...
- 1,947
6
votes
0
answers
155
views
Metric spaces containing a topological disc
It is well-known that every connected, locally connected compact metrizable space $X$ contains an arc, that is, a subspace homeomorphic to $[0,1]$. Are there topological properties we can add to these ...
- 7,193
6
votes
1
answer
261
views
When does base-change in topological spaces preserve quotient maps?
The question when $(-) \times X$ preserves colimits in topological spaces is well-studied. Since it always preserves arbitrary coproducts (disjoint unions), one only has to show when it preserves ...
- 12.1k
3
votes
0
answers
191
views
Does "Invariance of domain" hold true for injective Darboux function (instead of continuous injection)?
Let $f \colon U\subset \mathbb{R^n}\to\mathbb{R}^n$ be an injective Darboux map.
Does this imply that $f$ is an open map?
If $f$ is continuous then the result follows from "Invariance of domain&...
- 67
6
votes
1
answer
500
views
A characterization of metric spaces, isometric to subspaces of Euclidean spaces
I am looking for the reference to the following (surely known) characterization of metric spaces that embed into $\mathbb R^n$:
Theorem. Let $n$ be positive integer number. A metric space $X$ is ...
- 1,446
9
votes
2
answers
540
views
Can you fit a $G_\delta$ set between these two sets?
Every subset of $\mathbb N \times \mathbb N$ can be viewed as a relation on $\mathbb N$. The set $\mathcal P(\mathbb N \times \mathbb N)$ of all relations on $\mathbb N$ has a natural topology with ...
- 225
6
votes
1
answer
393
views
Algebra generated by transformation matrices
Let $T_n$ be the full transformation monoid of an $n$-set $N_n$ with elements 1,...,n consisting of all functions $f: N_n \rightarrow N_n$.
We can associate to each function $f$ a matrix $M_f$ in the ...
- 18.4k
5
votes
1
answer
152
views
Cartan matrix of the full transformation monoid ring
Let $T_n$ be the full transformation monoid of an $n$-set and $A_n=KT_n$ its monoid algebra over the complex numbers.
Question 1: Is the Cartan matrix of $A_n$ known? Im especially interested to see ...
- 26.5k
10
votes
1
answer
259
views
Space with compactly closed diagonal but which is not weak Hausdorff
Using the definitions from Peter May's A Concise Course in Algebraic Topology, a topological space $X$ is weak Hausdorff if for every compact Hausdorff space $K$ and continuous function $f:K\to X$, $f(...
- 41.9k
5
votes
1
answer
223
views
Is the interior of the tensor product of two convex cones equal to the tensor product of their respective interiors?
I am sorry that the following question is elementary. I have not received an answer from my post at Math Stack Exchange.
In the following question, all cones are convex and contain the origin. Let $C \...
- 6,406
53
votes
3
answers
8k
views
Grothendieck's manuscript on topology
Edit: Infos on the current state by Lieven Le Bruyn: http://www.neverendingbooks.org/grothendiecks-gribouillis
Edit: Just in case anyone still thinks that Grothendieck's unpublished manuscripts are (...
- 894
1
vote
0
answers
41
views
How to embed an arbitrary graph into (k,d)-kautz space (like multidimensional scaling of non-normed space)? See details in the following
How to embed an arbitrary graph into (k,d)-kautz space (like multidimensional scaling of non-normed space)? See details in the following.
Given a graph $G = \{V,E\}$,
we have a distance matrix (the ...
- 11
5
votes
2
answers
621
views
Image of the Hilbert space under a continuous bijection
Consider a continuous bijection $X\to Y$ such that $X$ is homeomorphic to a separable infinite dimensional Hilbert space. I wonder what can be said about topological properties of $Y$.
To exclude ...
- 29.1k
0
votes
0
answers
165
views
Topological property of an algebraic stack and its presentation
I started to learn algebraic stacks this January. I found there are several properties of algebraic stacks which are defined in terms of their underlying topological spaces, for example, connectedness,...
- 455
5
votes
1
answer
551
views
Under what conditions is the compact-open topology compactly generated?
Specifically, I'm wondering, if X and Y are Hausdorff, and Y is compactly generated, does it follow that C(X,Y), with the compact-open topology, is compactly generated?
Edit: answered as written, but ...
- 2,130
5
votes
2
answers
223
views
Continuous functions on $[0,1]^\omega$ and a product lower bound
I have a concrete question about continuous functions on $X = [0,1]^\omega$ (with the product topology).
The map $f:X\to [0, 1]$ given by $(x_i)\mapsto \prod x_i$ is well-defined and Borel but not ...
- 128k
4
votes
2
answers
702
views
Strongly zero-dimensional topological spaces and a simillar condition
A Hausdorff topological space $X$ is called strongly
zero-dimensional whenever for every closed subset $A$ of $X$ and
every open subset $U$ of $X$ such that $A \subseteq U$, there
exists a clopen ...
- 351
16
votes
2
answers
1k
views
Is there always a way up?
I am trying to find a simple criterion for a real continuous function $f$ on a connected, open subset $U$ of $\mathbb R^n$ that would imply the following property (P)
For any $x, y \in U$ such that $f(...
- 1,676
8
votes
2
answers
362
views
Is every contractible homogeneous space of a connected Lie group homeomorphic to a Euclidean space?
Problem. Let $G$ be a connected Lie group and $H$ is a closed subgroup of $G$ such that the homogeneous space $G/H$ is contractible. Is $G/H$ homeomorphic to a Euclidean space $\mathbb R^n$ for some $...
- 658
3
votes
1
answer
698
views
Simple proof that downward intersections of simply connected compact sets are simply connected
Let $X$ be a topological space, and $S_0, S_1, \dotsc \subset X$ be simply connected compact sets with $S_{n+1} \subset S_n$.
Question: Is there a simple proof that $S = \bigcap_n S_n$ is simply ...
- 12.9k
4
votes
1
answer
248
views
A question about regular closed sets
$\DeclareMathOperator\cl{cl}$Let $X$ be a topological space and let $Y$ be a dense subspace of $X$. Suppose
that $R\left( X\right) $ denotes all regular closed subsets of $X$.
Question 1: $R\left( Y\...
- 1,367