**3**

votes

**0**answers

160 views

### Every convex sequentially closed set is closed

Let $X$ be a vector space. A vector (not necessarily Hausdorff) topology on $X$ will be called convex sequential if every convex sequentially closed subset of $X$ is closed.
Is there some description ...

**1**

vote

**0**answers

82 views

### Tightening a loop

Consider two $d$-dimensional convex polytopes $c_1, c_2$ that share a $(d-1)$-dimensional face $f$. Let $M$ be a surface ($2$-manifold) that intersects each of $c_1$ and $c_2$ in a $2$-ball. Suppose ...

**4**

votes

**1**answer

184 views

### Sequentiality of largest vector topology

I know that the largest vector topology on countable dimensional vector space is sequential (i.e. every sequentially closed set is closed). Does it keep for the arbitrary vector space?
In countable ...

**12**

votes

**3**answers

261 views

### $T_2$ topologies that are “as disjoint as possible”

Let $X$ be an infinite set. Are there Hausdorff topologies $\tau_1, \tau_2$ on $X$ such that $\tau_1\cap\tau_2 = \{\emptyset\} \cup \{U\subseteq X: X\setminus U\text{ is finite}\}$?
(That is, the ...

**7**

votes

**2**answers

213 views

### Reconstructibility of topological spaces

Let $(X,\tau), (Y,\sigma)$ be topological spaces with $|X|$ infinite and suppose $\varphi:X\to Y$ is a bijection such that for all $x\in X$ we have that $(X\setminus\{x\}) \cong (Y\setminus\{\varphi(x)...

**0**

votes

**1**answer

46 views

### On 1-iso maps and subsets of the unit circle

Let $S$ be the unit circle and for any $x,y \in S$ let $d(x,y)$ be the lenght of the smallest arc between $x$ and $y$. A bijective map $\phi : S\longrightarrow S$ is called 1-iso if the following ...

**0**

votes

**1**answer

93 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\}\...

**2**

votes

**0**answers

47 views

### Existence of enough local sections

Let $\pi: G\to X$ be a continuous open (!) surjection of locally compact Hausdorff spaces. Assume that each fiber $G_x=\pi^{-1}(x)$, $x\in X$ carries a group structure making it a locally compact ...

**1**

vote

**1**answer

98 views

### Intersections of families of open sets ordered by well-inside relation in Euclidean space

Let $\langle\mathbf{R}^n,\mathscr{O}\rangle$ be the $n$-dimensional Euclidean space. Define $\mathbf{Q}\subseteq\mathcal{P}(\mathscr{O})$ to consist of all sets $\mathsf{Q}$ which simultanously ...

**4**

votes

**1**answer

122 views

### $C(X)$ as finitely generated $C^*$-algebra

Let $X$ be a Hausdorff space. Suppose that $C(X)$ (or $C_0(X)$) is a finitely generated $C^*$-algebra. What we can say about $X$ ? For example can we characterize its inductive dimension, axioms of ...

**6**

votes

**1**answer

294 views

### Classify $K(\pi,n)$ that are manifolds

Inspired by `Naturally occuring' $K(\pi, n)$ spaces, for $n \geq 2$. and When is a classifying space a topological manifold?, I'd like to formulate a precise question:
For which $n \in \mathbb{Z}...

**2**

votes

**0**answers

63 views

### Pre-cosheaf of connected components

Consider a continous map $f:Y \to X$ between topological spaces. The pre-cosheaf $\mathcal{F}: Open(X) \to Set$ of connected components of the inverse image is defined as $\mathcal{F}(U):= \pi_0(f^{-1}...

**4**

votes

**1**answer

156 views

### Is the topological dual of a Banach space weakly* closed in its algebraic dual?

The question is completely contained in the title :)
I can only add, that it is not difficult to give a counterexample for normed spaces, and also Banach-Steinhaus theorem implies the sequential ...

**10**

votes

**3**answers

548 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 ...

**3**

votes

**1**answer

109 views

### If $S$ is non-stationary in $[k]^{\omega}$ is there a choice-function on $S$ with bounded fibers?

Fodor's Lemma : When $k$ is a regular uncountable cardinal, and $T$ is a stationary subset of $k$, any regressive $f:T\to k$ has a fiber which is stationary in $k$. Corollary: $T$ is stationary in $...

**3**

votes

**1**answer

211 views

### When is the generalized Cantor space $\kappa$-compact?

My M.Sc. student has the following question, that I assume has an answer in the literature, and we are looking for references.
The generalized Cantor space is the space $2^\kappa$, with basic open ...

**4**

votes

**1**answer

120 views

### Is the following product-like space a Polish space?

Let $\mathcal{M}_1(\mathbb R)$ denote the space of Borel probability measures on $\mathbb R$. The space is a Polish space (a space which admits a complete, separable, metric) using, say the Levy-...

**12**

votes

**2**answers

261 views

### Term for a metric space for which the triangle inequality is strict?

Is there a standard term for a metric space in which $\rho(p,r) < \rho(p,q) + \rho(q,r)$ for any distinct $p$, $q$, $r$? Sort of the opposite of metric convexity.
For instance, a subset of ...

**1**

vote

**0**answers

167 views

### A functor on the category of rings, algebras or compact Hausdorff topological space

Assume that $R$ is a unital ring or a complex or real (Banach or $C^{*}$) algebra.
We define a relation $M$ on $R$ as follows: $$a\;M b \;\;\; \text{iff}\;\; a=xy,\;b=yx \;\; \text{for ...

**0**

votes

**0**answers

100 views

### Alternative representation of $C_c(X)$ as inductive limit

CORRECTION: As Simon Henry points out in the comments, there is a problem in the construction: the maps $\varphi_n$ are not necessarily linear.
Under some additional constraints on the space (e.g. $X$ ...

**2**

votes

**0**answers

211 views

### The theory of frames and locales as elementary topology [closed]

In What is elementary geometry? (pdf) Alfred Tarski defined elementary geometry to be
that part of Euclidean geometry which can be formulated and established without the help of any set-...

**0**

votes

**1**answer

78 views

### $C^{\infty}_{loc}$-convergence - right definition

Let $\Omega \subset \mathbb{R}^{n}$ be some open set. Let $f_{n},f\in C^{\infty}(\Omega)$. My question is: What does the following phrase mean? $f_{n}$ converges to $f$ in $C^{\infty}_{loc}(\Omega)$. ...

**2**

votes

**0**answers

90 views

### Centralizer of a dense subgroup in a maximal subgroup of a reductive group

I am looking for a reference to the following statement
"Let $G$ be a reductive algebraic group and $K$ a maximal compact subgroup of $G$. If $H$ is a dense subgroup in $K$, then the centralizer of $H$...

**1**

vote

**1**answer

85 views

### Identifying attractors in high dimensional dynamical sytems [closed]

I have a high dimensional dynamical system, and I was wondering if there is a method to identify the various attractors of the system i.e, a way of mapping the energy landscape?
I was thinking of a ...

**0**

votes

**0**answers

57 views

### Can a countable and connected space be hausdorf? [duplicate]

Can a countable and connected space be hausdorf? If not, can it be T1, i.e., every pair of points is topologically distinguishable and seperable?

**1**

vote

**1**answer

90 views

### Isotopy class of closed 2-ball embedded in R^3

My intuition tells me that any two topological embeddings of the closed 2-ball (aka unit disk) into $R^3$ are isotopic in $R^3$. Is this correct? Maybe easy to prove?
It seems like it should be easy ...

**4**

votes

**3**answers

161 views

### End points of continua

Whyburn (1942) defined an end point $x$ of a continuum $X$ to be any point having arbitrarily small neighborhoods each of whose boundaries contains a single point. Thus, he defines an end point ...

**6**

votes

**2**answers

206 views

### A non locally compact group of finite topological dimension?

Is there a topological group which is Hausdorff, first countable,
locally connected and has finite topological dimension, yet fails
to be locally compact?

**3**

votes

**0**answers

61 views

### Can approximately periodic functions be perturbed to periodic functions on a locally compact group?

Let $G$ be a locally compact group and $H\subset G$ a closed and cocompact subgroup. I wish to consider bounded continuous functions from $G$ to $\mathbb{C}$ that are periodic in the following strong ...

**2**

votes

**1**answer

164 views

### Holonomic splitting

I am reading the book "Introduction to the h-Principle" by Eliashberg and Mishachev. At the moment I try to understand the Section 1.7 Holonomic splitting on page 12 but without success. I do not ...

**4**

votes

**0**answers

216 views

### A generalized ellipse

We know that an ellipse is the locus of all point $z$ in the plane with $$|z-a|+|z-b|=\lambda$$
where $a,b$ are two given points in the plane and $\lambda$ is a constant.
Now we consider the ...

**7**

votes

**3**answers

259 views

### Numerical and topological density

Let $\mathbb{N}$ denote the set of positive integers, and let's say that $A\subseteq \mathbb{N}$ is numerically dense if $$\text{lim inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} = 1.$$
Is there a ...

**8**

votes

**1**answer

194 views

### Simply connected noncompact surfaces

Is there a theorem saying that any noncompact, simply connected topological surface is homeomorphic to the plane ? There seems to be many well-known results about the classification of compact ...

**9**

votes

**3**answers

551 views

### Collection of dense subsets as a “fingerprint” for Hausdorff topologies?

Let $(X,\tau)$ be a Hausdorff space and let ${\cal D}$ denote the collection of dense subsets of $(X,\tau)$. Is it possible that there is another Hausdorff topology $\tau_1 \neq \tau$ on $X$ such that ...

**4**

votes

**1**answer

112 views

### Completeness of Localizations of Completions of Commutative Rings

Let $R$ be an integral domain. Let $x,y\in R\setminus\{0\}$ be distinct. Let $\hat R$ be the $x$-adic completion of $R$ (the ring of all sequences $(r_n+Rx^n)_{n\ge0}$ where for $n\ge0$, $r_n\in R$ ...

**-3**

votes

**1**answer

176 views

### Homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1) [closed]

Can one construct homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1)?
If so, please show me how to construct it.

**1**

vote

**0**answers

82 views

### Does bounded and closed equal compact for sets of Borel probability measures?

Equip the space of Borel probability measures on a fixed closed subset X of the s-dimensional Euclidean space with the topology induced by weak convergence of probability measures. In this setting, ...

**14**

votes

**0**answers

328 views

### pizza lemma (topology)

given six real-analytic arcs in the unit disk $D$, each of which
connects the origin to a boundary point, and no two arcs meet anywhere except
at the origin, and the arcs meet at equal (60 degree) ...

**2**

votes

**1**answer

79 views

### Is there any workable internal characterization of zero-sets?

I am asking the question in a purely topological setting; a zero-set of some topological space $X$ is a subset which can be realized as the counterimage of a single point through a continuous real-...

**2**

votes

**1**answer

141 views

### Extremally disconnected spaces and a measure theoretic property

Suppose that $X$ is an extremally disconnected topological space (meaning that the closure of an open set is still open). Then $X$ has the following property: the family of all sets $S$ such that $S$ ...

**3**

votes

**0**answers

58 views

### Kind of multiplicative total boundedness in Hausdorff compact rings

Let $(R,\cal T)$ be a unital Hausdorff compact topological ring and let $A$ be an open subset of $R$ containing $1$. Is there a finite set $B$ with $AB=R$?

**2**

votes

**1**answer

107 views

### Two questions about the extent to which simple arcs and simple closed curves can fill up higher dimensional Euclidean spaces

For each positive integer n, let E(n) be n-dimensional Euclidean space with its standard metric and let p(n) be some fixed point of E(n). The so-called "Osgood Curve" shows that there can exist simple ...

**1**

vote

**0**answers

71 views

### Not normal connected component of a right topological group

Let $\cal T$ be a locally compact topology on a group $G$ and $(x,y)\mapsto xy^{-1}$ be continuous at $(1,1)$ and for every $a\in G$, $x\mapsto xa$ be continuous everywhere with respect to $\cal T$.
...

**2**

votes

**1**answer

63 views

### Regular open Boolean algebras and homomorphism which does not preserve nearness of sets

I am looking for an example of topological spaces $\langle X_1,\mathscr{O}_1\rangle$ and $\langle X_2,\mathscr{O}_2\rangle$ such that
there is a homomorphism $h\colon\mathrm{r}\mathscr{O}_1\...

**10**

votes

**1**answer

299 views

### Dimension in CW-approximation

The following question was something that came to my mind during my (unsuccessful) attempt at answering this MO-question.
Let $X$ be a topological space, and let $\tilde{X}\to X$ be a CW-...

**6**

votes

**1**answer

233 views

### Two definitions of Lebesgue covering dimension

Maybe this question has already been considered here, but after a quick search I didn't find what I was looking for.
As I see, in the literature there are two different definitions of the ...

**5**

votes

**1**answer

245 views

### Factorization of a certain map through a CW-complex

Suppose that $X$ is a paracompact Hausdorff space (e.g. a metric space) with $\dim X=n$ (the Lebesgue covering dimension). I want to find a proof (or a reference) that any (continuous) map $f: X \to K(...

**1**

vote

**0**answers

69 views

### Name for a type of weak path connectedness?

The following topological property arose in the context of my friend Don Hadwin's operator theory research, and he asked me to ask here if the property occurs in the literature and has a name.
...

**2**

votes

**1**answer

131 views

### Preservation of topological properties in between two topologies

Let $X$ be a set, $\tau_1 \leq \tau_2$ two comparable topologies on $X$ ($\tau_1$ is weaker than $\tau_2$) and consider some topological property $\varphi$ that holds for both $\tau_1$ and $\tau_2$.
I ...

**12**

votes

**0**answers

200 views

### Is there a locally compact, $\omega_1$-compact, not $\sigma$-countably compact space of size $\aleph_1$?

There are old ZFC examples due to Eric van Douwen that satisfy all the properties in the title, except that they are of cardinality $2^{\aleph_0}$, so the answer to the title question is YES if the ...