**1**

vote

**1**answer

88 views

### Banach Isomomorphic Cts Fucntion Algebras for two Non-Homeomorphic Top Spaces?

Can anyone provide me with an example of two non-homeomorphic locally-compact Hausdorff spaces $X$ and $Y$, such that $C(X)$ and $C(Y)$ are isomorphic as Banach algebras. Clearly, the ...

**2**

votes

**1**answer

225 views

### Dual of the space of continuous functions

Let $T \subseteq \mathbb R$ be a closed set of real numbers. Let $X := C(T, \mathbb R)$ denote the Fréchet space of continuous real-valued functions on $T$. The topology on $X$ is generated by ...

**5**

votes

**0**answers

138 views

### Series in topological rings that only converge if almost all summands are zero

While trying to understand a certain topological ring better, I stumbled onto the following question.
Suppose $I$ is a fixed infinite index set, $R$ is a topological ring and $(x_i)_{i\in I}$ is a ...

**4**

votes

**1**answer

345 views

### Topological Problems Solved by Lattice Duality

It is well known the success of lattice dualities (as Pontryagin duality for abelian groups, Stone duality for Boolean algebras and Priestley duality for distributive lattices) to solve algebraic ...

**12**

votes

**0**answers

223 views

### Connected and locally connected, but not path-connected

Allow me to use some non-standard terminology:
A h-contractible space is a non-empty topological space $X$ such that, for any topological space $T$ and any pair of continuous maps $f_0, f_1 : T \to ...

**3**

votes

**2**answers

167 views

### Product of Topological Measure Spaces

Def. A Radon measure $\mu$ on a compact Hausdorff space $X$ is uniformly regular if there is a countable family $\mathcal{A}$ of compact $G_\delta$-subsets of $X$ such that for every open set ...

**1**

vote

**0**answers

133 views

### Extending a homeomorphism from a dense set [closed]

Let $X$ and $Y$ be Hausdorff topological spaces, and let $f : X \to Y$ be a Borel-measurable function. Suppose that $D \subseteq X$ is dense, that the image $f(D) \subseteq Y$ is dense, and that $f$ ...

**4**

votes

**4**answers

628 views

### Categorical Construction of Quotient Topology?

The product topology is the categorical product, and the disjoint union topology is the categorical coproduct. But the arrows in the characteristic diagrams for the subspace and quotient topologies ...

**12**

votes

**2**answers

255 views

### If G is a sequential topological group, must G x G be sequential?

Using standard definitions, the topological space $Y$ is sequential if for each nonclosed $A \subset Y$, there exists a convergent sequence $a_{1}$ , $a_{2}$,...$\rightarrow b$
so that $a_{n} \in A$ ...

**1**

vote

**0**answers

137 views

### Can a compact metrizable space be determined by its Hausdorff measures?

Suppose that $(X,d)$ is a compact metric space. Now suppose that $h:[0,a]\rightarrow[0,b]$ is a continuous function with $h(0)=0$ where if $x\leq y$, then $h(x)\leq h(y)$. Then define ...

**0**

votes

**0**answers

85 views

### Analytic extension from a closed analytic subset

I have the following question: Let $\Omega \subset \mathbb{R}^{n}$ be an open set and consider $X \subset \Omega$ an analytic subset. By this I mean that there exists analytic functions ...

**2**

votes

**2**answers

184 views

### Tychonoff spaces and ideals

Let $X$ be a tychonoff space and let $T$ be the set of all $f \in C(X)$ such that for any $g$ the equation $fg = 1$ has at most finitely many solutions. Under what conditions on $X$, the set $T$ is an ...

**5**

votes

**2**answers

467 views

### CW complex and group action

This is a general question and any reference or related result will be extremely helpful.
Suppose $X$ is a Hausdorff topological space. Suppose G (a countable group) acts on it. Let $Y=X/G$ be the ...

**2**

votes

**2**answers

199 views

### A Fixed point Theorem that does not need the convexity of set valued map?

I am looking for a fixed point theorem for set valued maps that does not assume the set valued map should be convex valued.
Something like contractiblity or other properties can be replaced with ...

**3**

votes

**1**answer

156 views

### Extending the hyperbolic splitting on $\Lambda$ to a neighborhood of $\Lambda$

Let $M$ be a compact Riemannian manifold and let $f:M→M$ a diffeomorphism. Let $\Lambda\subset M$ be a compact invariant subset of $M$. We say that $\Lambda $ is a hyperbolic set for $f$ when there ...

**2**

votes

**2**answers

164 views

### A set intersecting the graph of any continuous function in a finite set

New version of the problem I am looking for a characterization of those completely regular and hausdorff spaces $X$ such that the follwing is true:
If $f :X\longrightarrow \Bbb{R}$ is continuous ...

**1**

vote

**1**answer

576 views

### Topological razors (ball-like spaces)

Introduction
Many admire the Euclidean space, and I am not an exception. I will try to catch the topological roundness of the $n$-ball in its greatest generality. I call the resulting axiomatized ...

**2**

votes

**1**answer

132 views

### Is there a maximal (or maximal Tychonoff) non normal space?

Is there a maximal (or maximal Tychonoff) non normal space? In "A Problem of Set-Teoretic Topology" the existence of a maximal Tychonoff space is asserted. Also there exists a perfectly normal maximal ...

**16**

votes

**2**answers

498 views

### Intersection of compact sets in the unit interval

Let $\mathscr K$ be an uncountable set such that every $K\in\mathscr K$ is a compact subset of $[0,1]$ with positive Lebesgue measure. Does it then follow that there exists an uncountable $\mathscr ...

**6**

votes

**2**answers

353 views

### “is topologically mixing” vs. “is topologically transitive” in the defition of chaos

This question is cross-posted from MSE, since it hasn't gotten an answer there for over 72 hours.
Wikipedia gives essentially "is topologically mixing and has dense periodic periodic orbits"
as the ...

**3**

votes

**3**answers

310 views

### A question concerning the isomorphic type of continuous functions

let $S$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are continuous outside a bounded open interval containing zero (depended on $f$). Is it possible to consider $S$ as ...

**0**

votes

**2**answers

162 views

### Is there a normal space that is not uniformly normal

Let $(X,\mathcal D)$ be a uniform space and $A,B\subseteq X$. Let's say $A$ is uniformly inside $B$ and write $A\le B$ iff there's some entourage $D$ for which
$$(\forall a\in A)(D[a]\subseteq B)$$
A ...

**1**

vote

**1**answer

80 views

### Generalized connected components decomposition for Priestley spaces

Preliminaries A partially ordered space is both a poset and a topological space. It has connected components both as a topological space, and connected components as a poset, i.e. the maximal ...

**0**

votes

**1**answer

216 views

### Which algebra of functions can be represented as $C(X)$

I don't know if this problem is known or not, so any information would be appreciated:
Let $\cal A$ be an $\Bbb{R}$-algebra of (not necessary continuous) real valued functions defined on a ...

**1**

vote

**1**answer

83 views

### Normal Uniform Spaces and their function uniform spaces

Let $(X,\mathcal D)$ be a normal (diagonal) uniform space and $G$ be the set of all homeomorphisms $f:X\to X$. Let $\Delta$ be the uniformity on $X^X$ (inherited by $G$) by subbase
$$\Lambda =\{ ...

**2**

votes

**1**answer

85 views

### Are the connected components of a Priestley space closed?

Preliminaries A Priestley space is both a poset and a topological space. The topologically connected components of the space are trivially closed. (They are just the points of the underlying set.) But ...

**5**

votes

**1**answer

190 views

### What's the link between topological spaces as locales and topological spaces as infinity-groupoids?

I've seen texts that talk about topological spaces being essentially locales, like Topology via Logic by Vickers, and texts related to homotopy theory that talk about topological spaces being ...

**5**

votes

**1**answer

576 views

### A problem on infinite dimensional metric space

Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces verifying :
$X_{n}$ is a regular CW complex of constant local dimension $n$.
$X_{n}$ is of finite type, ...

**1**

vote

**2**answers

152 views

### Metrization of spaces of functions

Let $M$ and $N$ be topological spaces. Are there necessary and sufficient conditions on the topological properties of the spaces such that $C(M,N)$ is metrizable?
For $M$ compact and $N$ a metric ...

**4**

votes

**1**answer

226 views

### Omitting types and Baire category

What is the relation between omitting types theorems in model theory and the baire category theorem?

**4**

votes

**2**answers

291 views

### Is there a Tychonoff space $X$ of cardinality not of the form $2^\alpha$ such that $|C(X)| = |X|$

Let $X$ be the real line with the usual topology. Then clearly $|C(X)| = c = |X|$ and on the other hand $|X| = 2^{\aleph_0}$.
Now my question is as in the title: Is there a Tychonoff space $X$ of ...

**0**

votes

**1**answer

301 views

### What are some characterizations of the strong and total variation convergence topologies on measures?

I asked this question on StackExchange a few days ago but didn't get any response, so I thought I would try here.
The Wikipedia article on convergence of measures defines three kinds of convergence: ...

**2**

votes

**1**answer

219 views

### A completely regular space that is very non-normal

Take a completely regular Hausdorff topological space $X$ considered as a subset of its Stone-Čech compactification $\beta X$. If $X$ is not normal, we can find a closed subset $Y$ of $X$ and a ...

**4**

votes

**1**answer

184 views

### Inducing metric spaces

Let $f\colon \mathbb{R}_{\geq0} \to \mathbb{R}_{\geq0}$ be a function. We say that $f$ has the property of inducing metric spaces, whenever for all metric space $(X,d)$, $(X, f \circ d)$ is also a ...

**1**

vote

**0**answers

84 views

### Proving that two given functionally structured spaces are isomorphic

The relevant definitions are listed below. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups; and Section 2, Chapter II of Bredon's Topology and ...

**10**

votes

**2**answers

381 views

### Are these rings of functions isomorphic?

Let $R$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are continuous outside $(-1,1)$ and let $S$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are ...

**5**

votes

**1**answer

201 views

### inverse problem for ergodic measures

It is a basic fact in the weak-* topology, the set of invariant measures for a dynamical system is closed, compact, and convex in the weak-* topology. Furthermore, the set of ergodic measures is equal ...

**8**

votes

**1**answer

310 views

### Partition $\Bbb{R}$ into a family of sets each one homeomorphic to the Cantor set

It is known that there is no (nontrivial) partition of $\Bbb{R}$ into a countable number of closed set. But is there a partition of $\Bbb{R}$ into sets, each one homeomorphic to the cantor ternary ...

**5**

votes

**1**answer

117 views

### Is There a maximal space that is a P-space?

we guess there is no maximal space which is also a P-space. Am I right? Do u know a counter example?
clarifications:
Maximal space is that space with topology $\tau$ which is maximal crowded topology ...

**6**

votes

**3**answers

474 views

### Continuous functions as uniformly continuous function

Three question concerninng metrics on the real line:
Is there a metric $d$ on $\Bbb{R}$ such that a function $f : (\Bbb{R},d) \longrightarrow (\Bbb{R},d)$ ( or $f : \Bbb{R} \longrightarrow ...

**5**

votes

**3**answers

163 views

### When are all continuous self-maps of a topological spaces generated by retractions and self-homeomorphisms of prime order?

This is probably a very basic question, but I don't know the answer and I also don't see how it might be obvious (which it very well might be).
Given a topological space $X$, when is the set of all ...

**2**

votes

**1**answer

338 views

### Different Metrics for Baire Space and their induced Topologies

The Baire-Space is the set of all infinite sequences of integers, i.e.
$$
\mathcal N = \omega^{\omega}.
$$
On this space usually the following metric is given
$$
d(\alpha, \beta) = \left\{ ...

**1**

vote

**1**answer

94 views

### countably normed spaces and countably normed spaces [closed]

Why locally convex spaces are not presented as countably normed spaces i.e an infinite sequence of norms (see Generalized functions Tome 2 by Gelfand and Chilov) in the western mathematical ...

**9**

votes

**0**answers

261 views

### Topology of marked groups for different number of generators

A $k$-marked groups is a pair $(G,S)$ where $G$ is a group and $S$ is an ordered set of $k$ generators of $G$. Each such pair can be identified with a normal subgroup of the free group $F_k$ of rank ...

**4**

votes

**0**answers

159 views

### Krull dimension and Morley rank

Definition : A Topological space $\mathcal{D}$ is called noetherian if it satisfies the descending chain condition for closed subsets. We define the dimension of $\mathcal{D}$ to be the supremum of ...

**-1**

votes

**1**answer

211 views

### Topological degree of homogeneous function of degree k [closed]

Let $F:\mathbb{C}\to \mathbb{C}$ be a homogeneous map of degree $k$ (i.e., $F(tx)=t^kF(x)$, $t>0$). It is true that $F$ has topological degree less than or equal to k?
This is true if F is ...

**1**

vote

**1**answer

338 views

### The space $\psi$

Is the space $\psi$ (described in problem 5I of L. Gillman and M. Jerison, Rings of continuous functions, Springer Verlag, 1976) a F-Z-space (i.e, space with $cl(X-Z(f))$ is a zero set for every $f$ ...

**6**

votes

**1**answer

241 views

### Formal series convergence in deformation quantization and $C^*$-condition

A link between formal series convergence in deformation quantization (strict deformation quantization) and producing $C^*$-algebras instead of mere $*$-algebras (which ...

**6**

votes

**0**answers

270 views

### Prokhorov's theorem for finite signed measures?

Prokhorov theorem provides a useful characterization of relatively compact sets w.r.t. narrow topology (topology induced by narrow convergence) in the space of probability measure.
Notation used ...

**3**

votes

**2**answers

265 views

### How to define compatible topology for first-order structures?

Background Because a bounded distributive lattice can be represented by the clopen sets of a Priestley space, I tried to learn some basics about Priestley spaces. After reading (on Wikipedia)
A ...