**5**

votes

**0**answers

106 views

### Topologically transitive, pointwise minimal systems

I'm cross-posting this from SE.
Let $T$ be a group, and let $(X,T)$ be a flow, i.e. $X$ is a compact Hausdorff space and $T$ acts on $X$ by homeomorphisms. A flow $X$ is called topologically ...

**2**

votes

**2**answers

259 views

### Are the closed and unbounded subsets of $\mathbb{R}$ known up to homeomorphism?

I am currently working on a problem for which this knowledge could greatly reduce the number of cases, but I have yet to find anything after searching online. Are the closed unbounded subsets of $\...

**2**

votes

**1**answer

77 views

### Sequentially indistinguishable topologies on a countable set

All of the famous examples for sequentially indistinguishable topologies on a set $X$ are provided on an uncountable set $X$ (an uncountable set $X$ with discrete and cocountable topology or the $l^1(\...

**2**

votes

**1**answer

177 views

### Relation between two different definitions for relative sequential compactness

Building upon this question in Math.SE, I think the following might be rather of interest for MO.
In the literature on measure theory, probability and functional analysis the definition of a subset $...

**3**

votes

**1**answer

151 views

### Invariants of category in Polish spaces

Consider the invariants appearing in the Cichoń's diagram: $add(\mathcal I)$, $cov(\mathcal I)$, $non(\mathcal I)$, $cof(\mathcal I)$, where $\mathcal I$ is either the ideal of null sets for the ...

**8**

votes

**0**answers

135 views

### Are there always large discrete families of normal measures?

Let $\kappa$ be a measurable cardinal. We give the Stone space of all ultrafilters on $\kappa$ the usual topology, where each $x\subseteq\kappa$ determines a basic open $[x]=\{U;x\in U\}$. The ...

**8**

votes

**1**answer

440 views

### How is Ricci flow related to computer graphics?

I recently came across the book Ricci Flow for Shape Analysis and Surface Registration: Theories, Algorithms and Applications by Wei Zeng and Xianfeng David Gu. Because, I just saw the book on the ...

**14**

votes

**2**answers

380 views

### Can an oriented closed $n(\geq 2)$-dimensional manifold be embedded in $\mathbb{R^{2n-1}}$

Can anyone provide me an example of an orientable closed manifold $M$ of dimension $n\geq 2$, which cannot be embedded in $\mathbb R^{2n-1}$?
I know this is certainly not true when $n=1$, i.e. $S^1$....

**1**

vote

**0**answers

66 views

### A $\mathcal{C}^1$ domain and Hausdorff dimension estimate

Let us consider an open connected domain $E\subset \mathbb{R}^N$ and fix a point $x_0\in\partial E$ on its boundary.
Suppose now, that there exists $R>0$ such that the set $\partial E \cap B_{R}(...

**0**

votes

**0**answers

38 views

### A $\mathcal{C}^1$ differentiable domain is $F_\sigma$?

Let us consider a domain $E\subset \mathbb{R}^N$ and fix a point $x_0\in\partial E$ on its boundary.
Suppose now, that for every $R>0$ the set $\partial E \cap B_{R}(x_0)$ is $\mathcal{C}^1$, i.e. ...

**2**

votes

**1**answer

78 views

### Construct a specific base for Fine uniformities in the diagonal(Entourages) case

For every uniformizable space $X$ there is a finest uniformity on $X$ compatible with the topology of $X$ called the fine uniformity or universal uniformity.
To construct Fine uniformities, Let ...

**8**

votes

**2**answers

288 views

### Maximal trivialising subspace for a vector bundle

Let $X$ be a locally compact Hausdorff space. Given a vector bundle $p: E\to X$, a subspace $Y$ of $X$ is called trivialising (for this bundle), if after restricting this bundle to $Y$, it is a ...

**2**

votes

**0**answers

203 views

### algebraic structure of Integral Steenrod squares

It is well known that the classical Steenrod squares $Sq^a$ satisfy the Adem relations
$$Sq^aSq^b= \sum_c \binom{b-c-1}{a-2c}Sq^{a+b-c}Sq^c\;.$$
In the case where $a$ is odd, one can define an ...

**2**

votes

**1**answer

110 views

### Is the interval topology of $(\mathbb{N}^\mathbb{N}, \leq^*)$ connected?

Given a quasi-ordered set $(Q,\leq)$ the interval topology on $Q$ is generated by
$$\{Q\setminus\downarrow x : x\in Q\} \cup \{Q\setminus\uparrow x : x\in Q\},$$
where $\downarrow x = \{y\in Q: y\leq ...

**-1**

votes

**1**answer

68 views

### Interval topology on $(\mathbb{N}^\mathbb{N},\leq^*)$

Given a quasi-ordered set $(Q,\leq)$ the interval topology on $Q$ is generated by
$$\{Q\setminus\downarrow x : x\in Q\} \cup \{Q\setminus\uparrow x : x\in Q\},$$
where $\downarrow x = \{y\in Q: y\leq ...

**7**

votes

**1**answer

300 views

### A property stronger than the fixed point property

Assume that $X$ is a topological space. We say that $X$ satisfies the strong fixed point property if the graph of every surjective continuous self-map intersect the graph of every continuous self-...

**1**

vote

**1**answer

101 views

### A weak fixed point property

The usual fixed point property can be interpreted in terms of non empty intersection of the graph of all maps with the graph of the identity map.
This motivates us to consider the following "weak ...

**1**

vote

**0**answers

90 views

### Is there another equivalence relation on based maps between spheres which form the same graded ring as the homotopy groups?

Let $\sim$ be an equivalence relation on continuous based maps from $S^k$ to $S^n$, where $k$ and $n$ range over the positive integers.
Suppose that
Given maps $f, f^\prime: S^k \to S^n$ and $g, g^...

**5**

votes

**0**answers

59 views

### Almost invariance in compact quotients of locally compact groups

While trying to get an analogue of Weiss's monotiling result for amenable residually finite groups
in the topological setting, I face the following problem.
Let $G$ be a locally compact amenable ...

**2**

votes

**1**answer

186 views

### Topologies for which the ensemble of probability measures is complete

I have been struggling quite a bit with reconciling my intuitive understanding of probability distributions with the weird properties that almost all topologies on probability distributions possess.
...

**8**

votes

**0**answers

138 views

### Topology of family of complex varieties

It seems to be an oft-cited fact (which comes up, for instance, in describing vanishing/nearby cycles) that:
For a proper flat map $f \colon X \rightarrow \Delta$, where
$X$ is a complex ...

**3**

votes

**1**answer

130 views

### Is it possible for a random nowhere dense closed set to have a positive probability of hitting any given point?

Given a compact metrisable topological space $X$, we write $\mathcal{N}(X)$ for the set of non-empty closed nowhere dense subsets of $X$, which is a Polish space under the topology induced by the ...

**2**

votes

**1**answer

168 views

### Is the set of entire functions Borel in the space of analytic functions?

$\def\bbR{\mathbb R}\def\bbC{\mathbb C}\def\scrT{\mathscr T}\def\ssp{\kern.4mm}
$More specifically, I ask whether $S$ be a Borel set in the topological space $(\Omega,\scrT)$ in the following ...

**12**

votes

**3**answers

292 views

### Is a certain subset of the disc a convex set?

Some one asked me this question and I thought about it and I don't have any good idea to solve that. Can some one help me and give me an idea to start solve that?
Draw a Cantor set $C$ on the circle ...

**8**

votes

**0**answers

142 views

### Which nice subcategories of $\mathsf{Top}$ are locally cartesian closed?

For a class $\mathcal{J}$ of topological spaces, let $\mathsf{Top}_\mathcal{J}$ denote the category of $\mathcal{J}$-generated spaces, i.e. those spaces $X$ such that $U\subseteq X$ is open iff $f^{-1}...

**1**

vote

**0**answers

228 views

### The closure of a set of closed points

Let $X $ be a compact non-Hausdorff topological space with the following property: for every infinite subset of closed points, say $\{x_i\}_{i \in I}$, there exists $j\in I$ such that $x_j$ is in the ...

**1**

vote

**1**answer

74 views

### The separated uniform space associated with $(X,\mathfrak{U})$

If $\mathfrak{U}$ is a not necessarily separated uniform structure for some set $X$, then an equivalence relation $R$ can be introduced on $X$ by letting $x R y$ provided $(x,y)\in U$ for every $U\in \...

**2**

votes

**1**answer

118 views

### Is the complement of the ends of a manifold bounded?

Let $M$ be a connected manifold with precisely $k$ ends $\epsilon_1,...,\epsilon_k$. Choose a collection $(U_i)_{i=1}^k$ of pairwise disjoint open $\epsilon_i$-neighborhoods. Then I wonder how to ...

**2**

votes

**1**answer

178 views

### Can we Characterise Rings of Continuous Functions?

Suppose $K$ is some nice space, for example $\mathbb R$ or $\mathbb C$. Let $X$ be a set and $C$ a ring of functions $X \to K$. Is there any way to determine, from the algebraic structure of $C$, ...

**2**

votes

**3**answers

263 views

### Classification of open subset of $\mathbb{R}^{3}$ [closed]

There is a theorem which gives a classification of connected open sets of $\mathbb{R}^{2}$. Unfortunately, I don't remember the correct statement, but it looks like this
Theorem ? Let $U\subset\...

**6**

votes

**1**answer

331 views

### all subsets borel

Assume Martin's axiom plus $\neg CH$. It is well known, via almost disjoint forcing, that every set of reals of size less than continuum is an example of a metric space whose subsets are all ...

**5**

votes

**1**answer

273 views

### Which Banach spaces are realcompact?

I have a question about the topological space underlying a Banach space.
A topological space $X$ is realcompact iff it is homeomorphic to a closed subset of an infinite product of the form $\mathbb R^...

**0**

votes

**1**answer

85 views

### Complement of a finite union of convex sets

Question. Let $V_1,\ldots,V_n$ be open, bounded and convex subsets of $\mathbb R^2$. Show that $F=\mathbb R^2\smallsetminus\bigcup_{i=1}^n V_i$ possesses only finitely many connected components.
I ...

**1**

vote

**2**answers

199 views

### How do I prove that compact-open topology is metrizable?

Let $X$ be a $\sigma$-compact topological space and $(Y,d)$ be a metric space.
Let $\{K_n\}$ be a sequence of compact subsets of $X$ whose union is $X$.
Define $\rho_n(f,g):=\sup \{d(f(z),g(z)): z\...

**2**

votes

**2**answers

209 views

### Existence of a continuous section

Let $f\colon X\to Y$ be a surjective continuous map between two topological spaces such that $X,Y$ are path-connected and such that every fibre $f^{-1}(y)$ is connected, for each $y\in Y$. Is there ...

**3**

votes

**1**answer

97 views

### Homotopy equivalence of maps with compact support and maps which vanish at infinity

Let $X$ be a nice space, maybe a manifold, and let $Y$ be a based space.
What sort of conditions must we impose on $Y$ (and $X$ if need be) to get a homotopy equivalence
$$
\mathcal C_c(X,Y) \simeq \...

**5**

votes

**0**answers

81 views

### Minimal Hausdorff topologies compatible with a bunch of functions

Let $X$ be an infinite set, let ${\cal F}$ be a set of functions $f: X\to X$. We say that a topology $\tau$ is compatible with ${\cal F}$ if every $f\in {\cal F}$ is a continuous function $f:(X, \tau)\...

**6**

votes

**2**answers

315 views

### Is any function taking compact sets to compact sets, and connected sets to connected sets, necessarily continuous? [closed]

It is well-known that continuous image of any compact set is compact, and that continuous image of any connected set is connected.
How far is the converse of the above statements true?
More precisely:...

**12**

votes

**0**answers

170 views

### Do the Laver tables converge to the Sierpinski triangle with a line segment sticking out in the hyperspace topology?

Let $(\{1,...,2^{n}\},*_{n})$ denote the $n$-th Laver table.
Let
$$C_{n}=\{(\frac{x}{2^{n}},\frac{x*_{n}y}{2^{n}})|x,y\in\{1,2,3,...,2^{n}\}\}$$
for all $n\in\mathbb{N}$.
Then since $C_{n}$ is a ...

**6**

votes

**2**answers

255 views

### The role of the index set in the product of uncountably many topological spaces

Let $\langle X_i,\mathcal{T}_i \rangle_{i\in I}$ be a family of topological spaces. Consider $X=\prod_{i\in I} X_i$ with product topology.
Question. Is there a topological property that holds in $...

**2**

votes

**1**answer

211 views

### Is it true that given any two point in $M$ if there exists an unique geodesic joining those two points, then $M \sim \mathbb{R^n}$ [closed]

This following doubt initially came to my mind while thinking the relationship between number of genus of a manifold and number of geodesic between given two points.
DOUBT: Suppose $M\subset \mathbb{...

**8**

votes

**1**answer

176 views

### Embedding abelian cancellative Hausdorff topological semigroups into abelian Hausdorff topological groups

An abelian cancellative semigroup embeds (via a semigroup monomorphism) into an abelian group. What about an abelian cancellative Hausdorff topological semigroup that does not embed (via a ...

**6**

votes

**2**answers

75 views

### Closed field lines in the plane

A dipole in the plane consists of a positive charge P and an equal and opposite negative charge N separated by a fixed distance . Almost all of the resulting electric field lines (which fill the ...

**1**

vote

**1**answer

195 views

### Extension of continuous and smooth functions

Let us consider any subset $U \subset \mathbb{R}^{n}$. By definition, a function $f: U \rightarrow \mathbb{R}^m$ is smooth if, for every $x \in U$, there exist an open neighbourhood $\Omega_{x}$ of $x$...

**6**

votes

**1**answer

98 views

### Countable subcover of half-open cylinders

While preparing a lecture on dynamic programming principle in optimal stochastic control after the book of Touzi, I discovered a gap in the proof of DPP (page 28 of the book).
Here I simplify the ...

**42**

votes

**3**answers

1k views

### Duality between Compactness and Hausdorffness

Consider a non-empty set $X$ and its complete lattice of topologies
(see also this thread).
The discrete topology is Hausdorff. Every topology that is finer than a Hausdorff topology is also ...

**3**

votes

**0**answers

81 views

### When closed subsets have finitely many connected componenets

Let $X$ be topological space such that every its closed subset has finitely many connected componenets. Is there any charactrization for such topological space?

**9**

votes

**1**answer

429 views

### Is Max (R) a Hausdorff space?

I asked the following question at < http://math.stackexchange.com/questions/1508168/is-max-r-a-hausdorff-space >, but I pose it here for any help.
Recall a space is totally disconnected if the ...

**2**

votes

**0**answers

114 views

### Notion of convergence on a dense subset

My motivation for this question is as follows.
Consider the set $D$ of cadlag functions on $(0,1)$ and its subset $D^\uparrow$ of non-decreasing cadlag functions.
Each $f \in D$ has at most countably ...

**5**

votes

**2**answers

165 views

### Existence of non-homeomorphic pair of bijectively related closed subsets in $\mathbb{R}$

I want to find two closed, non-homeomorphic subsets $A$ and $B$ of $\mathbb{R}$ (with subset topology), with the property that there exist two continuous bijections
$$f:A\to B,~~~~g:B\to A.$$
Clearly ...