All Questions
5,184 questions
0
votes
1
answer
360
views
Triviality of finite fiber bundles [closed]
Hello,
I suspect the following is true and easy but I am unable to prove. Suppose (E, B, π, F) is a fiber bundle, where E,B are compact and F is finite, prove that E is a trivial fiber bundle. Any ...
0
votes
1
answer
165
views
How many n-dimensional closed submanifolds of $R^n$ have Euler characteristic 1?
It is well-known that the closed $n$-ball has Euler characteristic $1$. Is it true that every closed (i.e., compact), connected $n$-dimensional submanifold (with boundary) of $\mathbb R^n$ having ...
0
votes
2
answers
545
views
Suppose $(G,\mathcal T)$ is a paratopological group and $a,b\in G$ and every neighborhood of $a$ contains $b$. Can we say every neighborhood of $b$ contains $a$?
Suppose $(G,\mathcal T)$ is a paratopological group and $a,b\in G$ and every neighborhood of $a$ contains $b$. Can we say every neighborhood of $b$ contains $a$?
clearly every closed neighborhood ...
user31967
0
votes
1
answer
403
views
When does a power semigroup have a zero, and what can the zero be?
Let $S$ be a semigroup. The power semigroup of $S$ is the set $P(S)=2^S\setminus\lbrace\varnothing\rbrace $ with the operation $$AB=\lbrace ab\ |\ a\in A,\ b\in B\rbrace.$$
This operation is ...
- 1,445
0
votes
2
answers
503
views
A Jordan arc in the unit disk
Let $D$ be the open unit disk, and $J$ a Jordan arc (that is, a homeomorphic copy of $[0, 1]$) that lies in $D$, except $J(0)$ lies on the boundary of $D$, say $J(0)=1$. I would like to see that $D\...
- 95
0
votes
1
answer
79
views
Dimension of a manifold derived from a dense $G_{\delta}$ subspace
Let $X,Y$ be (compact connected) topological manifolds of dimensions $n,m$, respectively. Assume that a dense $G_{\delta}$ subspace $A$ of $X$ is homeomorphic to a dense $G_{\delta}$ subspace $B$ of $...
0
votes
1
answer
192
views
A continuous injection from the Hilbert cube to the real line?
Continuing an earlier "too good to be true" question that I posted recently, the same holds for the present question:
Is there a continuous injection from the Hilbert cube $[0,1]^{\Bbb N}$ ...
- 3,104
0
votes
1
answer
525
views
Non-diffeomorphic but homeomorphic (under Lorentzian topology) Lorentzian manifolds
$\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\diff}{\mathrm{diff}}\newcommand{\manifold}{\mathrm{manifold}}$Take a time-oriented Lorentzian ...
- 612
0
votes
1
answer
157
views
Does there exist a star-Lindelöf space which is not star-$L$-Lindelöf?
A space $X$ is said to be star-Lindelöf if for every open cover $\mathcal U$ of $X$ there exists a countable subset $\mathcal V\subseteq\mathcal U$ such that $St(\cup\mathcal V,\mathcal U)=X$.
A ...
- 505
0
votes
1
answer
272
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
0
votes
1
answer
392
views
Set of zeros of a real function of class $C^1$ [closed]
Let $a < b$ two real numbers and let $f \colon [a,b] \to \mathbb{R}$ a $C^1$ function. Moreover, we consider the set
$$
X := \{ x \in [a,b]\mid f(x) = 0 \}.
$$
Is it the number of connected ...
- 29
0
votes
1
answer
177
views
Existence of certain continuous curves
Let $\mathbb{S}^n$ be the $n$-sphere. I would like to know if anyone knows of the following result in the literature (or whether anyone knows a proof/counterexample).
Let $f\colon\mathbb{S}^1\times[0,...
- 1,453
0
votes
2
answers
285
views
Motivation and reference for Brauer algebras
I am looking for a good reference and motivation for Brauer monoid and Brauer algebras. Kindly help me with some suggestions. Thanks.
- 141
0
votes
1
answer
228
views
Uniform distance from a discontinuous function is continuous
Define the metric $d(f,g)\triangleq \sup_{x \in [0,1]} \|f(x)-g(x)\|$ on the set $\operatorname{B}$ of uniformly bounded functions from the interval $[0,1]$ to $\mathbb{R}$, fix $g \in \operatorname{B}...
- 5,405
0
votes
1
answer
497
views
Separability of an algebra is equivalent to separability of its spectrum
Let $A$ be a commutative C*-algebra.
I would like to show that $A$ is separable (i.e. has a countable dense subset) if and only if the spectrum of $A$ (denoted by $\Omega(A)$) is separable.
Notes ...
- 143
0
votes
1
answer
283
views
Continuity of the Restriction Map Between Function Spaces [closed]
Let $X,Y,Z$ be Hausdorff spaces and suppose that $Z\subset X$. Endow $C(X,Y)$ and $C(Z,Y)$ with the compact-open topologies and define the map $\rho$ as
\begin{align}
\rho:&C(X,Y)\rightarrow C(Z,...
- 5,405
0
votes
3
answers
192
views
Connected $T_2$-spaces with only constant maps between them
If $f:\mathbb{R}\to\mathbb{Q}$ is continuous, then it is constant. Are there infinite connected $T_2$-spaces $X,Y$ such that the only continuous maps $f:X\to Y$ are the constant maps?
- 48.9k
0
votes
1
answer
233
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 ...
- 42k
0
votes
1
answer
638
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}^...
- 371
0
votes
1
answer
238
views
Minimal totally separated spaces
Let us call a space $(X,\tau)$ totally separated (ts) if for every two distinct points there is a clopen set containing one, but not the other. If for every topology $\sigma\subseteq\tau$ with $\sigma\...
- 48.9k
0
votes
1
answer
396
views
A Question about SO(n)
My question is:
How to find out all the finite subgroup of SO(n)? Or just for the simple case SO(4) SO(5)?
With more discribe:
If $S^n\backslash \Gamma$ is a manifold,
I just want to know that ...
- 703
0
votes
3
answers
424
views
Level 2 Menger Sponge
Hi fellows,
Does anyone know the number of holes of a level 2 Menger Sponge ?
user23855
0
votes
1
answer
224
views
Special functions on the unit disk
Let $\mathbb{D} = \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1 \}$ be the unit disk.
We say a function $f : \mathbb{D} \rightarrow \mathbb{D}$ is a winner if it satisfies the following:
1) it is a ...
- 1,271
0
votes
2
answers
370
views
zeroset-diagonal
Is it true that a topology space X with a zeroset diagonal is first countable?
what if X is additionally CCC?
- 654
0
votes
2
answers
909
views
Topology generated by the collection of open sets
Hello, there is a statement as following:
If every point of X is a G_delta and X is T_1, then take Y = set of X,
plus the topology generated by all open sets needed to prove G_delta-ness of every ...
- 654
0
votes
2
answers
478
views
"Exotic" Banach spaces of sequences
Does there exist a linear subspace of $\mathbb C ^{\mathbb N}$ that can be endowed a Banach space topology that is not finer than the locally convex topology of pointwise convergence?
Best,
Martin
- 5,327
0
votes
1
answer
386
views
The functor of continuous functions from compact CW-spaces to the reals
The contravariant functor $C(-)$ given by
$$
\hom_{Top}(-,\mathbb{R}):cCW\to Rng
$$
where $cCW$ is the category of compact CW complexes is injective on objects. What is known about surjectivity, ...
- 2,782
0
votes
1
answer
423
views
What Is This Quotient Space?
Let $X$ be a finite CW-complex with only even cells $x_1,\ldots, x_k$ and let $Y$ be the complex obtained by attaching one more even cell to $X$, call it $y$. Assume both $X$ and $Y$ are connected. ...
- 61
0
votes
3
answers
248
views
how slow can the dimension of a product set grow?
Let us define the following "dimension" of a Borel subet $B \subset \mathbb{R}^k$:
$\dim(B) = \min\{n \in \mathbb{N}: \exists K \subset \mathbb{R}^n, ~{\rm s.t.} ~ B \sim K\}$,
where $\sim$ denotes "...
- 1,839
0
votes
1
answer
67
views
Is there a characterization of monoids that distribute over each other?
Let $(M, e_1, \times_1, e_2, \times_2)$ be an algebraic structure such that
$(M, e_1, \times_1)$ and $(M, e_2, \times_2)$ are monoids
$x \times_1 (y \times_2 z) = (x \times_1 y) \times_2 (x \times_1 ...
- 631
0
votes
1
answer
100
views
Embeddings of pseudo metric spaces into seminormed Spaces
There is a theorem stating that every metric space embeds isometrically into $\ell _{\infty}$.
My question: is there a generalized result for pseudo metric spaces embedding isometrically into semi-...
- 85
0
votes
1
answer
328
views
Relationship between quotient CW-complexes after attaching cells
I have been trying to prove the following simple-looking result which I require for some work in low-dimensional topology. I expect it is likely true and in a textbook somewhere so any reference or ...
- 187
0
votes
1
answer
107
views
A question about the Stone-Čech compactification and ultrafilter
Let $X$ be a Tychonoff space and let $\beta X$ is the Stone-Čech
compactification of $X$. Assume $f:X\longrightarrow \mathbb{R}$ is a bounded function. Then there exists a function $f^{\beta }:\beta X\...
- 1,367
0
votes
1
answer
115
views
Generalized Triangle Inequality for Snowflakes
Let $p>0$ and consider a metric space $(X,d)$. I have recently come across a problem where the space $(X,d^q)$ provides is natural; where $q>1$. However, the triangle inquality break (i.e. it ...
- 169
0
votes
1
answer
517
views
Distance between two points using triangulation
Suppose we have two points $p_1$ and $p_2$ in a metric space with unknown dimensionality, with no way to directly compute the distance between them, e.g. no coordinates.
Say we can randomly sample a ...
0
votes
1
answer
504
views
Is every compact contractible subset of $\mathbb{R}^n$ homeomorphic to a closed ball of some dimension? [closed]
Question: Is every compact contractible subset of $\mathbb{R}^n$ homeomorphic to a closed ball of some dimension?
This post doesn't quite answer my question because it is about open sets.
- 637
0
votes
1
answer
249
views
About uniform continuity
Is there a definition Df(g) of uniform continuity of g, without using the notion of metric?
Let $(E,d_E)$ and $(F, d_F)$ metrics spaces, $f$ continuous fonction of $E$ to $F$
We must have :
Df$(f)$ ...
- 4,074
0
votes
1
answer
536
views
About the normability of the space of continuous functions
Let $A$ be a subset of $\mathbb{R}^n$, and denote by $C(A)$ the space of complex-valued continuous functions defined on $A$. We know that if $A$ is compact then we can define a norm on $C(A)$ so that ...
- 1,173
0
votes
1
answer
213
views
Hurewicz theorem on mappings that lower dimension
A form of Hurewicz theorem on mappings that lower dimension states that: Let $X$ and $Y$ be compact metric spaces and $f:X\to Y$ a continuous map. Suppose that there is some $n$ so that for every $y\...
- 675
0
votes
1
answer
188
views
Sober spaces vs. spatial frames-a big picture
For any topological space $X$ one can consider the so called frame of all open subsets of $X$ to be denoted by $\mathcal{O}(X)$. If $f:X \to Y$ is continuous taking the inverse image we get the ...
- 9,330
0
votes
2
answers
219
views
Intrinsically defining smooth/continuous/analytic functions
In mathematics, the notion of a continuous/smooth/analytic function $\mathbb{R}\to\mathbb{R}$ is introduced by defining the general set-theoretic function $\mathbb{R}\to\mathbb{R}$ and then imposing ...
user158636
0
votes
1
answer
120
views
Breaking up dense subset in non-separable space
Let $X$ be a not necessarily separable (infinite-dimensional) Banach space and $D\subseteq X$ be dense linearly independent subset. Then does there exist a set of infinite-dimensional separable ...
- 5,405
0
votes
1
answer
285
views
Explicit examples of (probability) measures on $\prod \mathbb{R}$
Let $\prod_{n \in \mathbb{N}} \mathbb{R}$ be equipped with the Tikhonov product of the Euclidean topologies on $\mathbb{R}$ and let $B$ the corresponding Borel $\sigma$-algebra. What is are some ...
- 5,405
0
votes
2
answers
344
views
subspace topology and strong topology
Suppose $X$ is a locally convex space and $Y$ is a subspace of the strong dual of $X$, is the induced topology on Y equivalent to the strong topology $b(Y,Y')$ on $Y$? If this is not correct, then on ...
- 71
0
votes
1
answer
282
views
Continuity of $\arg\min$
Let $f:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}$ be a continuous function.
Let $A = \{(x,\arg\min_x f(x,y)) | x \in \mathbb{R}^n\}$.
Is there necessarily a continuous function $g: \mathbb{R}^...
- 101
0
votes
1
answer
112
views
A category of topological spaces with certain anti symmetric property
Is there a category $\mathcal{A}$ of topological spaces, saturated with respect to homeomorphism relation, which is maximal with respect to the following property?
For every $X,Y \in \mathcal{A}...
- 356
0
votes
1
answer
165
views
Topology on $\omega\times\omega$ such that topologically connected equals graph-connected
For any set $X$ we define $[X]^2 =\big\{\{a,b\}: a, b \in X\text{ and }a\neq b\big\}$.
Let $$E = \big\{\{(a_1, a_2), (b_1, b_2)\}\in[\omega\times\omega]^2: |a_i-b_i| = 1\text{ for some } i\in\{1,2\}\...
- 48.9k
0
votes
1
answer
76
views
Countable, $T_1$, and not metacompact
Is there a countable space that is $T_1$ and not metacompact? (A space $(X,\tau)$ is not metacompact iff there is on open cover $\cal{U}_0$ such that for every open refinement $\cal V$ there is $x\in ...
- 48.9k
0
votes
2
answers
148
views
Continuous image relation on topological spaces
Let $\kappa$ be a cardinal, and let $\text{Top}(\kappa)$ be the set of topological spaces $(X,\tau)$ such that $X\subseteq \kappa$. We pre-order $\text{Top}(\kappa)$ by
for $X, Y \in \text{...
- 48.9k
0
votes
1
answer
128
views
What are the semigroups in which congruence classes can be multplied like sets?
For a semigroup $S$ and a congruence $\rho$ on $S$, let's say that $\rho$ is good when for all $a,b\in S$ we have that $[ab]=[a][b],$ where $[x]$ denotes the congruence class of $x$ modulo $\rho$ and ...
- 1,445