All Questions
5,184 questions
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
3
votes
1
answer
589
views
Extending open maps to Stone-Cech compactifications
(Cross posted from this math.SE question)
Let $X$ be a Cech-complete space, and $Y$ a paracompact space. Suppose $f\colon X\to Y$ is a continuous and open surjection.
Since $Y$ is completely ...
- 39.8k
3
votes
1
answer
529
views
Study of free monoids of the recursive S. Eilenberg.
Compared to the usual treatises on recursion (eg, Rogers H. "Computability and Undecidability." McGraw-Hill, New York) the book of Samuel Eilenberg & Calvin C. Elgot "Recursiveness" treats such ...
- 4,165
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
3
votes
1
answer
367
views
submonoid of a matrix monoid with a common eigenvector
Hello,
I am considering two real invertible $3\times 3$ matrices $A$ and $B$ and a nonzero vector $v\in\mathbb{R}^3$ and i am wondering if the submonoid $E$ of the monoid $(A,B)$ genererated by $A$ ...
- 69
8
votes
1
answer
802
views
A topological space for which having the ccc is independent of ZFC?
It is well known that a generalized Cantor space $2^A$ is separable if and only if $|A| \leq 2^{\aleph_0}$. This means that one cannot decide in $ZFC$ whether the space $2^{\omega_2}$ is separable or ...
- 11.5k
1
vote
1
answer
494
views
Is a section of a proper map proper?
Suppose $f\colon X \rightarrow Y$ is a continuous map of topological spaces and $s\colon Y \rightarrow X$ is a continuous section to $f$, i.e., $f\circ s = 1$. If $f$ is proper does this mean that $s$ ...
- 13
20
votes
2
answers
691
views
A "dimension" for Tychonoff spaces
It's well-known that any Tychonoff space $X$ can be embedded in $[0,1]^k$ for some cardinal $k$. It's natural to ask what the smallest such $k$ is (let's call it $k(X)$). However, this probably ...
- 301
15
votes
0
answers
1k
views
Is the category of smooth manifolds equivalent to the opposite category of the category of commutative monoids of some additive symmetric monoidal category?
This is a followup to my previous question, which asked whether
the category of commutative or noncommutative C*-algebras or von Neumann algebras
is equivalent to the category of commutative or ...
- 37.8k
8
votes
1
answer
365
views
Counting copies of a BA within a BA: arbitrarily many vs infinitely many
Informally, I am wondering if a Boolean algebra $\mathcal{B}$ contains infinitely many disjoint copies of a Boolean algebra $\mathcal{A}$ whenever it contains arbitrarily many disjoint copies of $\...
- 596
1
vote
3
answers
884
views
Does the manifold of the three dimensional group of rotations SO(3) cause a separation of space in the group of rigid motions SE(3)?
The group of three dimensional rotations $SO(3)$ is a subgroup of the Special Euclidean Group $SE(3) = \mathbb{R}^3 \rtimes SO(3)$. The manifold of $SO(3)$ is the three dimensional real projective ...
- 13
2
votes
1
answer
512
views
Question about analytic curves
Here a question that has me stumped. Maybe someone familiar with algebraic or differential curves can help. Suppose that $\gamma:[0,1] \rightarrow \mathbb{C}$ is an analytic function. Is it true ...
- 203
1
vote
1
answer
752
views
3D surfaces of infinite genus
How might one show that the set of connected 3D surfaces with infinite genus (up to homeomorphism) is countably infinite?
We could either use proof by contradiction or come up with a way to count ...
- 11
4
votes
0
answers
137
views
Does this property of scattered spaces have a name?
(Note: I asked this question at MSE a week ago and received no answer, so I am now reposting it here.)
Let $K$ be a (Hausdorff) scattered topological space and for each ordinal $\alpha$ denote by $K^{...
- 2,378
1
vote
1
answer
595
views
When is a bijective map between bundles a homeomorphism?
Let $F \rightarrow E_i \rightarrow X_i$ be a bundle with fibre $F$ for i=1,2.
Let $f:E_1 \rightarrow E_2$ be a bijective continuous map and $h: X_1 \rightarrow X_2$ a homeomorphism.
Is f then also ...
- 165
41
votes
4
answers
5k
views
Topological Characterisation of the real line.
What is a purely topological characterisation of the real line( standard topology)?
- 521
2
votes
2
answers
343
views
Action of centralizer on Borel-Moore homology of Springer Fibers for Affine Hecke Algebra
In Chriss and Ginzburg's "Representation Theory and Complex Geometry", they describe a geometric construction of representations of the affine Hecke algebra, using the Borel-Moore homology of ...
- 83
4
votes
1
answer
3k
views
Dense sets in the space of continuous functions
Let $X$ be a compact metric space, and
let $C(X)$ be the Banach space of continuous real-valued function on $X$, with
the maximum norm.
Suppose $S\subset C(X)$ is a set of functions with the ...
- 43
2
votes
1
answer
483
views
Whether fine topology and uniform topology on C(X,Y) coincide , when metric on Y is bounded
Whether fine topology and uniform topology on C(X,Y) coincide , when metric on Y is bounded
- 121
4
votes
1
answer
382
views
Is the subspace of DVR's of the Zariski-Riemann space still quasi-compact?
If $S$ is the Zariski-Riemann space of a noetherian subring $k$ of a field $K$, Zariski-Samuel prove that $S$ is quasi-compact. If $S'$ is the subspace of valuations that are discrete (i.e. that ...
- 1,347
0
votes
1
answer
493
views
Sheaf of sections and local triviality
This is probably not a research level question, I'm sorry if it is inappropriate. I'm reasking here this question on math.se.
Suppose that $\xi: E \to B$ is a bundle (by which I mean simply a ...
135
votes
5
answers
31k
views
Does the inverse function theorem hold for everywhere differentiable maps?
(This question was posed to me by a colleague; I was unable to answer it, so am posing it here instead.)
Let $f: {\bf R}^n \to {\bf R}^n$ be an everywhere differentiable map, and suppose that at each ...
- 114k
19
votes
4
answers
8k
views
Unique limits of sequences plus what implies Hausdorff?
It is known that there are non-Hausdorff spaces which admit unique limits for all convergent sequence (see here) and it is also known that unique limits for nets implies Hausdorff.
What I am ...
- 12.7k
6
votes
1
answer
555
views
Is there a name for the class of metric spaces such that the closure of the open ball of radius $r$ around each point $x$ is the set of elements $y$ such that $d(x,y)\leq r$ ?
Let $(X,d)$ be a metric space, let $B(x,r)$ be the open ball of radius $r$ about $x$ and $N(x,r)$ be the set of elements $y\in X$ such that $d(x,y)\leq r$. It is well-known that it is not always true ...
- 6,034
15
votes
4
answers
734
views
Continuously selecting elements from unordered pairs
The symmetric square of a topological space $X$ is obtained from the usual square $X^2$ by identifying pairs of symmetric points $(x_1,x_2)$ and $(x_2,x_1)$. Thus, elements of the symmetric square can ...
- 44.4k
3
votes
1
answer
353
views
Topological space with some conditions
Can one give an example of non-compact space $X$ which satisfies the following conditions:
the countable union of compact subsets is relatively compact,
for every closed noncompact subset $A$ of $X$ ...
- 145
0
votes
1
answer
296
views
homeomorphism of topological group
Let G be a topological group and a be an element of order 2 in G. Further suppose the element a does not belong to center of G. Then is it true that only homeomorphism f of G such that $f(ax)=f(x)a$ ...
- 1
4
votes
1
answer
354
views
Does the weak approximation theorem hold for general topological fields?
The weak approximation theorem states that given a field $F$ and nontrivial inequivalent absolute values $|\cdot|_1,\ldots,|\cdot|_n,$ and letting $F_i$ denote $F$ with the topology from $|\cdot|_i$, ...
- 2,585
13
votes
5
answers
5k
views
Is the preimage of the closure the closure of the preimage under a quotient map?
Let $f : X \to X/\sim$ be a quotient map from a topological space $X$ to the quotient space $X/\sim$ for $\sim$ some equivalence relation. Let $S \subseteq X/\sim$. Is it true that $f^{-1}(\overline{S}...
- 3,290
0
votes
1
answer
2k
views
What does it mean to have Zero Density (mathimatically) [closed]
I read a question that asked "prove that the set of all positive integers expressible as the sum of two integers square has zero density." Now I was under the impression that something was dense iff ...
- 93
70
votes
28
answers
7k
views
Examples where it's useful to know that a mathematical object belongs to some family of objects
For an expository piece I'm writing, it would be useful to have good examples of the following phenomenon:
(1) ${\cal X}$ is a parameterized family of somethings. (Varieties, schemes, manifolds, ...
0
votes
0
answers
179
views
semigroup actions of groups on regular rooted trees
If $G$ is a group which has a semigroup action on a regular rooted tree via prefix-preserving, continuous transformations (I give the tree the path metric), what kinds of algebraic restrictions can we ...
- 125
5
votes
2
answers
4k
views
finite codimension implies closed?
Let $E$ be a (complete) topological vector space, and $u:E\to E$ be continuous. Is it always true that if ${\rm Im}(u)$ is of finite codimension in $E$, then it is closed in $E$ or do we have to ...
- 53
2
votes
0
answers
1k
views
Double Torus Parametric Surface [closed]
In the process of trying to find continuous parametric surface equations for the double torus and for a pair of pants, I believe that the problem is unsolvable for some topological reason.
I have ...
3
votes
4
answers
514
views
Better terminology than "equivalence class of functions"
Let $X = C(\mathbb R)$ be the Fréchet space of real-valued continuous functions. For each $f \in X$ and each compact set $D \subseteq \mathbb R$, let $$[f]_D = \{ g \in X : \mbox{$g(t) = f(t)$ for ...
- 8,512
4
votes
2
answers
2k
views
Injective Function on a Dense Set
This is a topological question that came up tangentially to some material I was working on. Suppose $X$ and $Y$ are complete metric spaces and $D$ is a dense subset of $X$. Let $f:D\mapsto Y$ be a ...
- 445
6
votes
1
answer
623
views
When is the cofibrant replacement of a product the product of the cofibrant replacements?
I'm in a situation where I'd like to prove $Q(E\otimes E) \simeq QE \otimes QE$ for a monoid $E$ in a symmetric monoidal model category. I know it's not true in general that $Q(E\otimes F)\simeq QE \...
- 30.3k
7
votes
1
answer
796
views
Disconnecting sets
If E is a metric space, I call a subset C of E a cut if E-C is not connected and if C is minimal for this property (which is obviously equivalent to "for every p in C, E-C union p is connected". The ...
- 3,630
7
votes
2
answers
2k
views
Product of ultrafilters, is it an ultrafilter?
Let $a$ and $b$ are filters. The product $a\times b$ is defined as the filter (on the set of pairs) induced by the base $\{ A\times B | A\in a, B\in b \}$.
It is simple to show that product of a non-...
- 765
3
votes
0
answers
877
views
The "pullback presheaf" and the proper base change theorem in topology
Let $f:X\rightarrow Y$ be a continuous map of topological spaces and let $\mathcal{F}$
be a sheaf (say of abelian groups to fix the idea) on $Y$. Define the following rule on open sets of $X$:
$$
V\...
- 7,609
7
votes
2
answers
598
views
A characterisation of well-ordering ?
It is easy to prove that if $E$ is well-ordered, and if $f$ is a strictly increasing map from $E$ to $E$, then, for all $x$ in $E$, $f(x) \ge x$ (just consider the sequence $x$, $f(x)$, $f(f(x))\dots$)...
- 3,630
61
votes
1
answer
5k
views
Every real function has a dense set on which its restriction is continuous
The title says it all: if $f\colon \mathbb{R} \to \mathbb{R}$ is any real function, there exists a dense subset $D$ of $\mathbb{R}$ such that $f|_D$ is continuous.
Or so I'm told, but this leaves me ...
- 32.5k
1
vote
2
answers
412
views
When can the one-one continuous image of a perfect set fail to be perfect?
Let $\mathfrak{M}$ and $\mathfrak{N}$ be perfect Polish spaces, $P$ a nonempty perfect subset of $\mathfrak{M}$, and $f: \mathfrak{M} \rightarrow \mathfrak{N}$ a continuous surjection that's injective ...
- 1,081
-12
votes
1
answer
2k
views
Direct product of filters
Product $a\times b$ of filters $a$ and $b$ is defined as the filter (on the set of binary relations) defined by the base $\{ A\times B | A\in a,B\in b \}$.
I will denote the principal filter ...
- 765
2
votes
1
answer
987
views
A question about measurable structures on function spaces
Hey, I was just wondering, I'm using some of Robert Aumann's ideas about measurable structures on function spaces (From his paper 'Borel structures for Function spaces': http://projecteuclid.org/...
- 456
1
vote
1
answer
260
views
The intersection of Block Groups and R-trivial (finite) monoids
Let $\textbf{BG}$ be the pseudovariety of block groups, also known as $\textbf{EJ}, \textbf{PG},\ldots,\text{etc.}$(see [1]), and let $\textbf{R}$ be the pseudovariety of R-trivial monoids, by the ...
- 424
-9
votes
1
answer
2k
views
Filters and intersection of two binary relations
Let $\mathfrak{F}$ is the complete lattice of filters (including the improper filter) on some set, ordered
inverse to set-theoretic inclusion.
I will denote $\left\langle f \right\rangle \mathcal{X} =...
- 765
1
vote
1
answer
324
views
Sufficient conditions for Hausdorffness
Let $(X,\tau)$ be a $T_1$ topological space and $Y\subset X$ a dense subspace which is completely metrizable. Are there any sufficient conditions to ensure that $(X,\tau)$ is Hausdorff using the known ...
- 159
8
votes
1
answer
621
views
Sober except not $T_0$?
tl;dr: Is there an accepted or proposed term for a topological space whose $T_0$ quotient is sober?
The condition that a topological space be sober (and therefore equivalent to a locale) may be ...
- 2,754
2
votes
2
answers
809
views
On a special case of Alexander duality
Let $S^n$ be the $n$-dimensional sphere and let $K\subseteq S^n$ be a compact, locally contractible subspace of real codimension $\geq 2$. Applying Alexander duality we find that
$$
\tilde{H}_{i}(S^n-...
- 7,609