Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

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6
votes
1answer
174 views

What is $Sq^i(\alpha^j)$ for all $i$ and $j$?

Write $H^*(\mathbb{R}P^\infty; \mathbb{Z}_2) = \mathbb{Z}_2[\alpha]$, $\deg \alpha = 1$. What is $Sq^i(\alpha^j)$ for all $i$ and $j$? I am not an algebraic topologist by trade but need to know this ...
1
vote
0answers
180 views

Existence of topology on the space of continuous functions

Let $C:=C([0,1],\mathbb{R})$ be the space of real-valued continuous functions defined on $[0,1]$. Could we find a topological vector space topology $\pi$ on $C$ such that the following two conditions ...
1
vote
2answers
79 views

Is an open subset of a compact subset of a Hausdorff locally convex TVS paracompact?

This repeats the title in a more readable way. Take a compact subset $X$ of a Hausdorff locally convex topological vector space and $U$ be an open subset of $X$. Is $U$ paracompact?
5
votes
2answers
190 views

Does a continuous map $f$ from the $n$-ball $B$ into $R^n$ such that $B\subset f(B)$ have a fixed point?

If $f$ is a continuous map from the $n$-ball $B$ into itself, the Brouwer fixed point theorem guarantees a fixed point. What if we assume that $f$ maps $B$ into all $R^n$, and $f(B)$ contains $B$? For ...
6
votes
1answer
131 views

(A kind of) Irreducibiliy of regular open convex sets in the Cartesian space

I am looking for a proof of the fact which is formulated at the bottom of this post. The property of regular convex sets which the fact expresses seems to be true to me, yet I have not been able to ...
30
votes
3answers
1k views

Is there a subset of the plane that meets every line in two open intervals?

Using the Axiom of Choice, it is possible to construct a subset of the plane that meets every line in two points (these are called "$2$-point sets"). What if, instead of points, we ask for two open ...
6
votes
2answers
621 views

Is there a topological Chevalley-Shephard-Todd Theorem?

Is the following true: For a representation of a finite group $G$ on $\mathbb{C}^n$, the quotient $\mathbb{C}^n/G$ is a topological manifold if and only if $G$ is generated by pseudo-reflections. ( ...
10
votes
2answers
190 views

Concrete examples of covering from the 3-torus to the 3-sphere

There is a two-fold branched covering from 2-torus to the 2-sphere, $T^2 \rightarrow S^2$, whose covering transformation group is generated by the map $x \mapsto -x$ (Note that $T^2$ is an abelian ...
7
votes
1answer
207 views

Does “$\forall Z(C(X,Z) \cong C(Y,Z))$” imply $X\cong Y$?

If $X, Y$ are topological spaces, let $C(X,Y)$ denote the collection of continuous maps $f: X\to Y$, endowed with the compact-open topology. Assume that we are given topological spaces $X,Y$ such ...
2
votes
1answer
83 views

A quasicompact space with a net that contains no convergent strict subnet

If $x:\Lambda \rightarrow X$ is a net in a topological space $X$ and $\Lambda '\subseteq \Lambda$ is a cofinal subset of the directed set $\Lambda$, then $x|_{\Lambda '}$ is a subnet of $x$. We call ...
6
votes
0answers
185 views
+50

Thom Class of tensor bundles

Suppose $\xi$ and $\eta$ are oriented vector bundles over a CW-complex $B$. Is it possible to express the Thom class (with ${\mathbb Z}$ coefficients) of $\xi\otimes \eta$ or even ${\rm Sym}^2(\xi)$ ...
8
votes
2answers
213 views

Coarsest admissible topology on $\text{Cont}(X,Y)$

Let $X, Y$ be topological spaces and let $\text{Cont}(X,Y)$ be the collection of continuous functions $f:X\to Y$. We say that a topology $\tau$ on $\text{Cont}(X,Y)$ is admissible if the evaluation ...
4
votes
2answers
321 views

Is a specific sequentially closed subset of $M([0,1])$ closed?

Let $M([0,1])$ be the set of finite signed measures on $[0,1]$ (with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all ...
4
votes
4answers
250 views

When is the boundary of an open planar set a Jordan curve?

Is the boundary of an open, regular, bounded, path-connected, and simply connected set a Jordan curve? Trying to find weakest condition on an open bounded set to apply Carathéodory's theorem. My ...
3
votes
2answers
93 views

Given $x$ in a path-connected open set $S$ on the plane, are there non-crossing paths from $x$ to every point in $\partial S$?

A have a (non-simply) bounded path-connected open set $S\subset\mathbb{R}^2$. Given $x\in S$, there are paths in $S$ from $x$ to any point in the boundary $\partial S$. However, can all these paths ...
2
votes
1answer
39 views

Compact $R_1$-spaces

A space $(X,\tau)$ is said to be $R_1$ if for all $x,y\in X$ with $cl(\{x\}) \neq cl(\{y\})$, there are disjoint open sents separating $cl(\{x\})$ and $cl(\{y\})$. If $X$ is compact and $R_1$, does ...
1
vote
0answers
33 views

Decomposition which is locally connected

It is possible construct a connected compact metric space $X$ and a continuous decomposition $\mathcal{G}$ of $X$ that satisfies: 1)$X/\mathcal{G}$ is locally connected. 2)If $M$ is a compact ...
3
votes
1answer
183 views

Largest symmetric matrix given rank

Let $\mathscr{M}[n,d]$ be collection of $n\times n$ symmetric matrices with real entries from $\{0,1\}$ such that every row/column is distinct with sum of every row/column being $d$. What is minimum ...
4
votes
2answers
207 views

Sets $X,Y \subset [0,1]$, stronger than being measure $0$, such that $X+Y = [0,2]$

A set $X\subset \mathbb{R}$ is called nice if for every $\epsilon > 0$ there are a positive integer $k$ and $k$ bounded intervals $I_1,I_2,...,I_k$ such that $X \subset I_1 \cup I_2 \cup ...
2
votes
1answer
45 views

If $X$ has the “discrete” covering property, how about $X^2$?

We say that a space $X$ has covering property (C) if the following holds: (C) For any open cover ${\cal U}$ of $X$ there is a closed discrete set $D\subseteq X$ and a map $\varphi: D\to {\cal U}$ ...
3
votes
1answer
64 views

Can any $n$ dimensional (smooth, PL, topological) closed manifold be covered by $2^n$ pieces of $n$ dimensional real spaces?

For any $n$ dimensional closed manifold $M^n$, can we find an open covering $\{U_i\}_{i\in[2^n]}$ such that $M=\cup U_i$ and each $U_i\cong \mathbb R^n$? How about complex manifolds (replacing ...
3
votes
1answer
59 views

Example of a collection of metacompact spaces with non-metacompact box-product

Is there an example of a family $(X_i)_{i\in I}$ of metacompact spaces, such that their box product $\prod_{i\in I}^{\textrm{Box}}X_i$ is not metacompact?
0
votes
0answers
191 views

A question about the Leray-Serre spectral sequence

Suppose $F \to E \stackrel{p}{\to} B$ is a fibration with $B$ simply connected. The $E_2^{p,q}$ page of the Leray-Serre spectral sequence is given by $H^p(B;H^q(F))$. Suppose futhermore that $k$ is a ...
4
votes
1answer
158 views

Implications between different covering properties of spaces

Let $X$ be a set. A set ${\cal C}\subseteq {\cal P}(X)$ is said to be a cover of $X$ if $\bigcup {\cal C} = X$ and $X\notin {\cal C}$. If ${\frak U}$ and $\frak{W}$ are collections of covers of a ...
0
votes
0answers
60 views

Minimum rank of certain matrices

Let $\mathscr{M}[n]$ be collection of $n\times n$ matrices with real entries from $\{0,1\}$ such that every row is distinct and every column is distinct. What is minimum real rank of matrices in ...
2
votes
1answer
177 views

Embedding uncountably many disjoint copies of the Cantor set in the interval [closed]

Is it possible to embed uncountably many copies of the Cantor set in the unit interval so that any two are disjoint?
2
votes
1answer
213 views

A homeomorphism between total spaces with same fiber and base spaces not homotopic

Is there a counterexample to the following assertion?: Let $p_1:E_1\to B_1$ and $p_2:E_2\to B_2$ be fibrations with the same fiber $\mathbb S ^1$ such that $E_1$ and $E_2$ are homeomorphic (and both ...
1
vote
1answer
181 views

Class of functions between $C^{\infty}$ and $C^{\omega}$

I am always curious about that whether there exists a class of function which seems that more smooth than the $C^{\infty}$ class, while it is far from $C^{\omega}$ analytic function . From my point ...
5
votes
3answers
208 views

Which topological properties are preserved under taking box products?

Although the box topology is a topology worth studying and is similar to the strong topology in differential topology, the box topology is in many regards very badly behaved since the box product of ...
6
votes
1answer
158 views

Is there a universal $\omega$-limit set?

For the purposes of this question, a dynamical system means a compact metric space $X$ together with a continuous map $f: X \to X$. For $x \in X$, the $\omega$-limit set of $x$, denoted $\omega(x)$, ...
4
votes
0answers
110 views

Space of continuous real-valued functions on $[0,1]^\omega$ with the weak and pointwise topology

Let $C([0,1]^\omega)$ denote the set of continuous functions $f:[0,1]^\omega \to \mathbb{R}$. We endow $C([0,1]^\omega)$ with two topologies. Let $\tau$ be the pointwise topology on $C([0,1]^\omega)$ ...
2
votes
0answers
89 views

$T_2$-space $X$ with $X\cong \text{Aut}(X)$

Is there an infinite $T_2$-space $X$ with $X\cong \text{Aut}(X)$? (Here, $\text{Aut}(X)$ is the set of automorphisms $\varphi:X\to X$ and it carries the topology inherited from the product topology on ...
13
votes
1answer
442 views

Avoiding countable subgroups of a group homeomorphic to the Cantor space

The following question is motivated by the paper [Brian, Mislove, Every compact group can have a non-measurable subgroup]. A positive solution to a variation of the following problem implies a ...
3
votes
1answer
173 views

Does the CGWH-fication change the (weak) homotopy type?

Let $Top$, $CG$, $WH$, $CGWH$ be the categories of topological spaces, compactly generated spaces, weak Hausdorff spaces and compactly generated weak Hausdorff spaces. There is the CG-ification ...
3
votes
0answers
66 views

Are maps homotopic with respect to a uniform number of local homotopies

I've encountered the following problem that I'm sure someone more topologically inclined can answer: Say that a homotopy of maps $f:X\times[0,1)\to Y$ between two compact smooth manifolds $X$ and $Y$ ...
0
votes
1answer
98 views

Borel subsets of Polish groups

Suppose that I have a polish group $G$ and two subsets $A$ and $B$ of $G$ such that: $A$ is open in $G$ and $B$ is closed in $G,$ from this, can I conclude that $AB$ is a Borel subset of $G$? if not, ...
9
votes
0answers
131 views

A compact T1 topological space has a proper dense subset to which it is homeomorphic. What can be said about the space?

Let $X$ be a compact T1 (so singleton subsets are closed) topological space. Suppose that there is a proper subset $D \subset X$ such that: $D$ is dense in $X$; $D$ is homeomorphic to $X$. Note ...
4
votes
2answers
186 views

Exponential rule for Whitney-$\mathcal{C}^{\infty}$-topology

Let $M,N,X$ be smooth manifolds. Equip the space of smooth functions between two manifolds with the (strong) Whitney- $\mathcal{C}^\infty$-topology. The evaluation map $$ev\colon ...
11
votes
1answer
310 views

Non meager rectangle

Suppose $G \subseteq \mathbb{R}^2$ is dense $G_\delta$. Must there (in ZFC) exist non meager sets of reals $A, B$ such that $A \times B \subseteq G$?
4
votes
2answers
212 views

Is the space of signed finite measures on a compact set $M([0,1])$ a sequential space?

Let $M([0,1])$ be the set of finite signed measures on $[0,1]$ (with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all ...
7
votes
2answers
249 views

Space $X$ such that $X^\lambda\cong X$ for some $\lambda$

Which cardinals $\lambda > 2$ have the following property? There is a space $(X,\tau)$ such that for all cardinals $\kappa$ with $1<\kappa<\lambda$ we have $X\not\cong X^\kappa$, and ...
2
votes
0answers
100 views

Normed space that is sigma-totally-bounded but is not sigma-compact

Q1: Is there a separable normed space that is not sigma-compact, but is a countable union of totally bounded closed subsets? A test case is the space $C^1(I)$ with the $C^0$ norm where ...
-2
votes
1answer
85 views

Can every non-discrete topological group G be algebraically generated by a nowhere dense subset?

Is there somone help me to show that if this problem have positive Answer : Problem :Can every non-discrete topological group G be algebraically gen- erated by a nowhere dense subset ? Thank ...
6
votes
2answers
220 views

Spaces that can't be embedded in the plane

If $X$ and $Y$ are topological spaces, let us write $X \preceq Y$ whenever $X$ embeds in $Y$. Earlier today, I asked the question: Is this a well-quasi-order on the completely metrizable spaces? ...
5
votes
0answers
116 views

Homogeneous $\omega$-monolithic compact space

Under CH, is the cardinality of every homogeneous $\omega$-monolithic compact space $X$ not greater than $2^{\omega}$?
11
votes
3answers
516 views

The size of Lindelof space

Question. Suppose that $X$ is a Lindelof space such that every point of $X$ is a $G_{\delta}$-point. Then is it true that $|X| ≤ 2^{\omega}$?
1
vote
0answers
74 views

Monadicity of profinite algebras

We can show that the category of profinite algebras, cofiltered limits of finite algebras, is monadic over Stone spaces as follows. So, I wonder if there are any other examples. In case that I was ...
11
votes
1answer
202 views

Does Peano's existence theorem admits a constructive proof?

$$y(t)=y_0+\int_0^t b(y(s))ds$$ $b\in C(R^d)\cap L^\infty(R^d)$ The classical proof for Peano's existence theorem in ODE need use the Ascoli's theorem, so it's not constructive. When $d=1$, in the ...
7
votes
0answers
133 views

Intersection of connected components in $\mathbb{R}^n$

Let $n$ be a positive integer and let $K\subseteq \mathbb{R}^n$ be compact. Pick $x^* \in \mathbb{R}^n\setminus K$. Let $E$ be the connected component of $\mathbb{R}^n\setminus K$ that contains ...
1
vote
1answer
75 views

Problem about the existence of a continuous surjective map [closed]

Let $F$ be a closed set in $\Bbb R^2$, $F\neq \varnothing,\Bbb R^2$, and $F^\circ\neq \varnothing$, does there exist a continuous surjective map from $\Bbb R\times \Bbb Z$ to $F^{\circ-}-F^\circ$?