All Questions
5,185 questions
7
votes
1
answer
899
views
Is a closed subset of an extremally disconnected set again extremally disconnected?
Let $T$ be a compact Hausdorff extremally disconnected set (so $T$ is a compact Hausdorff space, such that the closure of each open subset is again open). Let $S \subseteq T$ be a closed subset.
...
CommunityBot
- 1
6
votes
1
answer
246
views
Is the projectivization of a topological vector space Tychonoff?
Let $E$ be a locally convex topological vector space over $\mathbb{R}$. The projectivization $PE$ is the quotient of $E\backslash\{0_{E}\}$ with respect to the equivalence relation $e\sim f$ if $e=\...
- 42k
13
votes
1
answer
1k
views
For what sets $X$ do there exist a pair of functions from $X$ to $X$ with the identity being the only function that commutes with both?
It is not too difficult to show that if $X$ is an infinite set, then there exists a two-element subset of the group $\operatorname{Sym}(X)$ with trivial centralizer iff $\lvert X\rvert \leq \lvert\...
- 32.5k
4
votes
0
answers
181
views
are trivial fibrations of finite CW-complexes soft for normal maps?
Are trivial Hurewicz fibrations of finite CW-complexes soft for normal maps,
i.e. is it true that for any trivial Hurewicz fibration $f:Y_1\to Y_2$
and a closed subset $A$ of a hereditary normal space ...
- 30.3k
10
votes
1
answer
394
views
Maximally continuous extension of continuous functions from $\mathbb Q$ to $\mathbb R$
Let $f: \mathbb Q \to \mathbb R$ be a continuous function.
An extension of $f$ is a function $\tilde f: \mathbb R \to \mathbb R$ such that $\tilde f = f$ on $\mathbb Q$.
We say an extension $\tilde f$ ...
- 12.6k
0
votes
0
answers
74
views
Do adjoining basepoints and/or moduli of spaces affect fixed points nicely?
My question is when will $(X_+)^G$ or $(X/A)^G$ be equal to $(X^G)_+$ or $X^G/A^G$ respectively for $X$ a $G$-space, $G$ a finite cyclic group and $X^G$ the ordinary fixed points. These seem like they ...
- 12.9k
40
votes
2
answers
2k
views
Can the nth projective space be covered by n charts?
That is, is there an open cover of $\mathbb{R}P^n$ by $n$ sets homeomorphic to $\mathbb{R}^n$?
I came up with this question a few years ago and I´ve thought about it from time to time, but I haven´t ...
- 35.9k
1
vote
0
answers
34
views
selection theory for normal non-paracompact domains?
Are there theorems in selection theory without either paracompactness or convexity assumptions ?
That is, a theorem that claims existence of selections for any (perfectly or hereditary) normal spaces, ...
- 317
3
votes
0
answers
80
views
Every Borel linearly independent set has Borel linear hull (reference?)
I am looking for a reference to the following fact, which probably is known and could be proved somewhere by someone.
Theorem. The linear hull of any linearly independent Borel set in a Polish ...
- 63.9k
2
votes
0
answers
74
views
Is the reversibility of inflation of a subset equivalent to its smoothness?
$D_r(x)$ denotes a closed ball of radius $r$ centered at $x$.
Definition. Let $M \subset \mathbb{R}^n$.
$D_r (M): = \bigcup\limits_{x \in M} D_r (x)$
$Int_r (M): = \{x ~|~ D_r(x) \subset M\}$
...
- 6,962
2
votes
0
answers
62
views
Does absolute retract imply convex structure?
In the theory of selection, it is known that any compact absolute retract (AR) carries a convexity structure defined by E. Michael. It is also known that a convex structure
developed by Van de Vel ...
- 2,083
10
votes
2
answers
843
views
Implicit function theorem with continuous dependence on parameter
Let $X,Y$ be Hilbert spaces and $P$ a topological space$^1$ and $p_0\in P$.
Let $f:X\times P\to Y$ be a continuous map such that
for any parameter $p\in P$, $f_p:= f|_{X\times \{p\}}:X\to Y$ is ...
- 2,388
4
votes
0
answers
156
views
Equivalence of definitions of Hirsch and Wall of strong $C^r$-topologies
I've been reading about strong (and weak) $C^r$-topologies on the space of $C^r$-maps between $C^s$-manifolds $M$ and $N$ ($s \ge r$) from the textbooks of Hirsch and Wall (both called Differential ...
- 63.9k
0
votes
0
answers
41
views
Selectively countable Boolean algebras of sets (terminology)
I am interested in the name for the following property of a Boolean algebra $\mathcal A$ of subsets of a set $X$:
$(\star)$ for any sequence $(A_n)_{n\in\omega}$ of pairwise disjoint nonempty sets in $...
- 42k
13
votes
3
answers
2k
views
The "miracle" of Heegard Floer.
Taking tori in symmetric products and "miraculously" proving that the Floer homology is independent of choices always seemed, well, miraculous. Some time ago Max Lipyanski explained to me the origins ...
- 35.5k
2
votes
0
answers
58
views
A generalization of metrics taking values in partial orders
I'm investigating the origin of the following notion:
Let $S=(S, +, <, 0)$ be a partially ordered semigroup with minimum $0$ (such that $<$ is invariant by the action of $+$ on both sides).
A $S$...
- 775
2
votes
0
answers
201
views
are acyclic fibrations of nice spaces absolute extensors for perfectly normal spaces?
A space $Y$ is called an absolute extensor for normal spaces (also sometimes solid) if, for any normal space $X$, closed subset $A$ of $X$, and map $f:A\to Y$, there exists a map $f′:X\to Y$ such that ...
- 317
10
votes
0
answers
272
views
What is the smallest $\sigma$-algebra of reals that is closed under addition of sets?
What is the smallest $\sigma$-algebra $\Sigma\subseteq\mathcal P(\Bbb R)$ containing the open sets and such that if $A,B\in\Sigma$, then $$A+B=\{a+b\mid a\in A,b\in B\}\in\Sigma?$$
I know that neither ...
- 3,011
1
vote
1
answer
153
views
Spaces whose interiors of retracts is a base of the topology
Definition: topological space $\ X\ $ is r-basic $\ \Leftarrow:\Rightarrow\ $ the interiors of retracts of $\ X\ $ form a topological base of $\ X.$
Main question: Are r-basic spaces mentioned in ...
- 6,962
2
votes
0
answers
73
views
Nonzero idempotents in compact semitopological semigroups with zero
Let $S$ be a compact semitopological semigroup. Then, $S$ contains minimal idempotents by Ellis' theorem.
Ellis' Theorem: In a compact left-topological (resp. right-topological) semigroup, every ...
- 2,605
9
votes
1
answer
255
views
Homotopy type of the Hausdorff metric
Recall that if we have a metric space $X$ then we can consider the set of its nonempty compact subsets and equip this with a metric called the Hausdorff distance. Denote the resulting metric space $\...
- 1,869
2
votes
1
answer
229
views
Baire 1 function equivalence in measure
I am trying to prove (or disprove) the following assertion:
Consider a probability triple $(X,\mathcal{B},\mu)$, $X$ separable Banach space (complete), $\mathcal{B}$ the Borel $\sigma-$algebra and $\...
- 32.5k
1
vote
0
answers
150
views
Lifting theorem for finite spaces: replacing perfect normality by normality
In the Lifting theorem for finite spaces (Thm. 3.5, Eric Wofsey, quoted below),
can one relax the condition "$A$ is a closed subset of a perfectly normal $X$" to
"$A\to X$ has the right ...
- 317
35
votes
13
answers
24k
views
Examples of non-metrizable spaces
I want to know some examples of topological spaces which are not metrizable. Of course one can construct a lot of such spaces but what I am looking for really is spaces which are important in other ...
- 7,067
8
votes
1
answer
442
views
Topological group locally homeomorphic to the Hilbert cube
Does there exist a topological group which is locally homeomorphic to the Hilbert cube $[0,1]^{\mathbb N}$?
Let me note that Hilbert cube has the fixed point property and thus it is not homeomorphic ...
- 63.9k
2
votes
0
answers
145
views
Semigroup ideals of a ring or an algebra
Let $R$ be a ring or an algebra. Suppose $B\subseteq R$ satisfies the property $BR \subseteq B$ and $RB\subseteq B$. Is there a general theory of subsets with this property in a ring (resp. algebra) ...
- 63.9k
24
votes
9
answers
2k
views
Self-containing structures
This question is partly inspired by this question: independently of the original context, I'm interested in the general claim* that an ill-founded set theory would represent certain mathematical ...
- 63.9k
3
votes
2
answers
509
views
Can the loops in the definition of the fundamental group be considered injective?
Let $\mathrm{С}$ be some class of topological spaces that includes at least all subspaces of $\mathbb{R}^n $. Further we are in the category $\mathrm{С}_{*}$ (the category of point spaces; all ...
- 12.9k
4
votes
0
answers
251
views
Topological field isomorphic to $\mathbb{C}$
Let $K$ be a topological field. If $K$ is a connected Hausdorff space, and is algebraically closed, is it true that $K$ is isomorphic to $\mathbb{C}$ ?
(I have deleted my question on MathStackExchange)...
- 353
2
votes
0
answers
119
views
The "matrix direct sum" monoid modulo unitary equivalence
Given a commutative $*$-ring $(R,*)$, let $M(R,*)$ be the monoid whose elements are matrices over $R$ of all possible shapes and entries, including those that have $0$ columns or $0$ rows. Let the ...
- 4,943
2
votes
0
answers
84
views
On the (Brouwer-Koebe) Continuity Method
The so-called Continuity Method is a simple yet powerful method to show that a given continuous injective map is surjective. Namely, suppose that $f:X \to Y$ is a map between two connected manifolds $...
- 2,154
7
votes
0
answers
194
views
Factoring a function from a finite set to itself
Let $S$ be a finite set and $f: S \to S$ be a function. Let $k = |f(S)|$ and let $\alpha$ be the partition of $S$ into $f$-fibers, i.e. $\alpha = \{ \alpha_t \}_{t \in f(S)}$ where $\alpha_t = f^{-1}(\...
- 695
1
vote
0
answers
70
views
Distance to set defined as subzero level set of a continuous function
I am searching for strategies on how to prove/disprove that scalar functions "capture" the distance to the subzero level set of the same function. (Or what topics to study to become better ...
CommunityBot
- 1
4
votes
1
answer
446
views
What is a "cusp" ("кусок") in relation to Guba's embedding theorem?
I'm confused by the definition of a "cusp" as found in
V.S. Guba, Conditions for the embeddability of semigroups into groups, Math. Notes 56 (1994), Nos. 1-2, 763-769 (link).
In the words of Mark ...
- 10.5k
-1
votes
1
answer
231
views
The set of prime numbers as a subspace of the Cantor set
We define an embedding of the set of prim numbers into the Cantor set as follows:
First we recall that the cantor set $\mathcal{C}$ is homeomorphic to $(\mathbb{Z}/10\mathbb{Z})^\omega $ since the ...
- 105k
30
votes
4
answers
2k
views
is f a polynomial provided that it is "partially" smooth?
Let $f$ be a $C^\infty$ function on $(c,d)$ ,and
let $O=\cup_{n\in \mathbb{Z}^+} (a_n,b_n)$ where $(a_n,b_n)$ are disjoint open interval in $(c,d)$ and $O$ is dense in $(c,d)$.
Suppose for each $n\in ...
- 60.6k
-1
votes
1
answer
116
views
Continuous surjection of $\mathbb{R}^{n-1}$ onto the interior of the $n$-simplex with continuous right inverse
Let $n$ be a positive integer. Clearly $\mathbb{R}^{n-1}$ and the interior of the $n$-simplex $
\delta_n := \{x \in [0,1]^n:\,\Sigma_k x_n =1, (\forall i)\,x_i>0\}
$ are homeomorphic. What I'm ...
- 63.9k
8
votes
1
answer
524
views
What is the status of Jordan's theorem in constructive mathematics in the language of locales?
By constructive mathematics in this matter we mean intuitionistic ZF (*).
In the language of locales, the Jordan curve can be defined as $f\colon S^1 \to X$ such that "if $U \cap V = \varnothing$,...
- 42.4k
29
votes
1
answer
812
views
Running most of the time in a connected set
Let $P$ be a compact connected set in the plane and $x,y\in P$.
Is it always possible to connect $x$ to $y$ by a path $\gamma$ such that the length of $\gamma\backslash P$ is arbitrary small?
...
- 50.6k
9
votes
2
answers
274
views
A differentiable one-parameter family of codimension 2 subspaces of $\mathbb{C}^n$ cannot fill $\mathbb{C}^n$, right?
Suppose that $P(t)$ is a one-parameter family of rank 2 self-adjoint projections on $\mathbb{C}^n$ that vary analytically in the real parameter $t \in [0,1]$. I claim that there must exist a vector $x ...
- 50.6k
2
votes
1
answer
783
views
Strong topology on a topological vector space
I'm not sure this is an appropriate question for this site but I've tried math stack exchange and got no answers. Also, this problem arose in one of my research problems, so I'm stating it here.
The ...
3
votes
1
answer
157
views
Is it true that the structure of a commutative ordered semiring is unique on a commutative ordered monoid?
Is it true that the structure of a commutative ordered semiring with identity is unique on a commutative ordered monoid (i.e., the structure of the monoid and the order are consistent)? I am not ...
- 18.7k
8
votes
1
answer
2k
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Is this question about US spaces still an open problem?
In Between $T_1$ and $T_2$, Albert Wilansky mentioned in 6. that it was not known whether or not every locally compact US space is $T_2$.
Is this matter still an open problem?
- 11.4k
1
vote
2
answers
223
views
Is $C_b(Q,E)$ linearly isometrically isomorphic to $C(\beta Q,E)$ where $\beta Q$ is the Stone–Čech compactification of $Q$?
Let $Q$ be a locally compact Hausdorff space and $E$ be a Banach space.
Let $C(Q)$ be the collection of all real-valued continuous functions on $Q$ and $C_b(Q,E)$ be the collection of all $E$-valued ...
- 4,713
1
vote
1
answer
182
views
Uniform covering and uniform continuity
Following
About uniform continuity
Let $E$ be a topological space, for all $a \in E$, we associate an open set of $E$, $U(a)$ containing $a$.
We will say that $\{U (a), a \in E\}$ is a uniform ...
- 39.8k
0
votes
1
answer
273
views
When do cobordism groups depend on differential structure? [closed]
I heard that cobordism group with structures sometimes depend on differential structure of space.
Do you know any examples or references about this facts?
I want to know when difference occur between ...
- 1
2
votes
1
answer
218
views
How to characterize the images of disk-algebra functions?
It is well known that the continuous images $f:\mathbf D\to \mathbb C$ of the closed unit disk $\mathbf D$ are exactly the non-void
compact, connected, locally path connected sets in $\mathbb C$.
...
- 91.8k
18
votes
3
answers
3k
views
What are parabolic bundles good for?
The question speaks for itself, but here is more details: Vector bundles are easy to motivate for students; they come up because one is trying to do "linear algebra on spaces". How does one motivate ...
- 4,713
0
votes
0
answers
75
views
Is second countability an extension property for non-Hausdorff spaces?
Let $G$ be an abelian topological group and let $H$ be a non-Hausdorff closed subgroup (so that $G/H$ is Hausdorff). If $H$ and $G/H$ are second countable, is $G$ second countable?
1
vote
1
answer
203
views
Open images of submetrizable spaces
In [Tka] the author writes:
"Every topological space $X$ can be represented as an open continuous image of a completely regular submetrizable space $Y$ (in other words, $Y$ admits a continuous ...
- 4,713