# Tagged Questions

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### First-countable topological monoids without local absorbing elements whose topology is induced by a semimetric

This is a follow up of Question 163246. For the reader's convenience, let me first copy&paste some basic definitions.
We let a semimetric on a set $X$ be a function $d: X \times X \to [0,\infty]$ ...

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145 views

### If $(\mathbb M, \tau)$ is a topological monoid, is $\tau$ always induced by a [left] subinvariant semimetric?

Let me start by recalling some basic definitions (just for the sake of avoiding misunderstandings due to the vocabulary of the post).
Basically following some ideas of W. Lawvere (but not his ...

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55 views

### the validity of a basic statement involving the Hausdorff distance

Let $\Omega_1 \supset \Omega_2 \supset \cdots$ be a sequence of nonempty, open, bounded and convex sets in $R^n.$
Define $\Omega = \operatorname{int} \Bigl( \overline{\bigcap_{k=1}^{\infty} \Omega_k ...

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753 views

### Who first defined _simply connected_, reference?

The following definition is due to Donald J. Newman:
A connected open subset $D$ of the plane $\mathbb C$
is simply connected
if and only if its complement $\widetilde D = \mathbb C \setminus D$
...

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117 views

### Approximation of continuous functions by Lipschitz functions in the topology of uniform convergence on compact sets

I was involved into this subject when I answered
this
question from MSE. Trying to generalize my answer, I am thinking about a following
Question. Let $X$ and $Y$ be metric spaces. When each ...

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**1**answer

93 views

### Projective limit and connected components

Let $E$ be a topological space. Let $\mathcal{K}$ be the set of the compact subsets of $E$.
$(E-K)_{K \in \mathcal{K}}$ is a projective system, because if $K,K'$ are two compacts, there are two ...

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**1**answer

146 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 ...

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178 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 ...

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149 views

### A few standard results (on metrizability and relative separation strength) without Choice?

I've been going back over some results from Munkres's Topology, and I'm curious about some things. (I originally posted this on M.SE, but I think it is probably a better fit here.)
I know that Choice ...

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153 views

### Finite non-Hausdorff models of CW-complexes

Years ago, my advisor showed me a construction where you take a CW complex and quotient each open cell to a single point. He said that under certain conditions (I believe always satisfied by the ...

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301 views

### Characterization of Fréchet-Urysohn spaces using sequential continuity at a point

A map $f \colon X \to Y$ is called sequentially continuous at the point $a$ if for every sequence $(x_n)$ such that $x_n\to a$, we also have $f(x_n)\to f(a)$.
$$x_n\to a \qquad \Rightarrow \qquad ...

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330 views

### Are there “chain complexes” and “homology groups” taking values in pairs of topological spaces?

Throughout this question, notation of the form $(X,A)$ denotes a sufficiently nice pair of topological spaces. I think for most of what I'm saying here, it is enough to assume that the inclusion $A ...

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**1**answer

342 views

### “monotone” homotopy?

This is a question about a concept that I call "monotone homotopy" which arises in a natural way in some topological situations.
Let $X$ be a (bounded) metric space, $Y$ be a topological space and ...

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**1**answer

265 views

### Riesz representation theorem for vector-valued fields

Let $Q$ be a locally compact Hausdorff space, and let $V$ be a topological vector space. Consider the space $X = C_0(Q, V)$ of $V$-valued fields which vanish at infinity. Let $X^*$ denote the dual ...

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**1**answer

361 views

### Is there a compact space with no countably generated dense subspace?

This is a reformulation of this MO question which recieved little or no attention due to the fact that the OP gave no motivation whatsoever. I found the question quite interesting and decided to give ...

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151 views

### When is a $*$-homomorphism between multiplier algebras strictly continuous?

(This question was posted on MSE here but didn't get any answers.)
The strict topology on the multiplier algebra M(A) of a C*-algebra A is that generated by the seminorms
$$ x\mapsto \|ax\|\quad ...

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612 views

### reference for “X compact <=> C_b(X) separable” (X metric space)

I know (and am able to prove via Stone-Čech compactification) that the following is correct:
Theorem: A metric space is compact if and only if its space of bounded, continuous, real-valued ...

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**1**answer

113 views

### perfect space without convergent long sequences

Is there a boolean space $X$ without isolated points with the property that no point $x\in X$ is the limit of a long sequence $(x_i)_{i\in I}$ from $X\setminus \lbrace x\rbrace $ ('long sequence' here ...

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**3**answers

288 views

### How should one look at the set of compatible ring structures on a given group?

Earlier today I had a conversation with a friend about ways of putting topologies on sets of first-order structures; we wound up talking about reducts and expansions from a topological point of view, ...

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232 views

### Descriptive set theory on $\mathbb{R}^\mathbb{N}$

The short version of my question is, What is a good source for learning about descriptive set theory on the space $\mathbb{R}^\mathbb{N}$, under the product topology coming from the discrete topology ...

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254 views

### Is the generalized Baire space complete?

I want to see whether the fact that the Baire space $\omega^\omega$ is a complete (metrizable) space generalizes to $\kappa^\kappa$ being a complete (topological) space. I think this is an easy ...

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248 views

### A continuous notion of realizability

I have been interested in non-classical logics, off and on, for quite a while. This question is probably very basic, and I hope it is not too low-level for MO. My question stems from an attempt to ...

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369 views

### C*-algebras with no nontrivial endomorphisms

Pick a C*-algebra $A$ and call a (*-)endomorphism $\alpha:A\to A$ nontrivial if it is injective and $\alpha(A)\neq A$.
Question: Do there exist infinite dimensional C*-algebras with no nontrivial ...

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**1**answer

250 views

### product spaces of rationals

Let $Q$ follow subspace topology from $R$
Then I think it is true that $Q^n$ and $Q^m$ (with product topology) are not homeomorphic.I also think it will be possible to define "rational" homotopy ...

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**1**answer

130 views

### How (and when) to factor a function defined on a product of metric spaces?

Suppose we have a set of regular functions defined on a product of metric spaces, for instance the Banach space of the smooth functions from $\mathbb R^n$ to $\mathbb C$. We know, thanks to the Taylor ...

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331 views

### Determining continuous functions on Banach spaces

Let $X$ be a real Banach space.
For a continuous (not necessarily linear) function $g:X \to \mathbb{R}$ and a family $\mathcal{F} \subseteq X^*$, we´ll say that $\mathcal{F}$ determines $g$ if ...

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345 views

### countably complete filters

Is there any description of the set of countably complete filters on the lattice of dense $G_{\delta}$ subsets of a compact, second countable metric space? [I haven't just dreamt this up: it describes ...

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1k views

### Where can I find a proof of the de Rham-Weil theorem?

Where can I find a proof of the de Rham-Weil theorem?
Does anyone know?

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324 views

### When is a Topological pushout also a Smooth pushout?

I feel like this problem has not been solved, but I'm interested in knowing any results on it. More specifically, I mean:
Let $B\stackrel{f}{\leftarrow} A \stackrel{g}{\rightarrow} C$ be a diagram ...

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246 views

### Constructing the Stone Space of a Distributive Lattice

Does anyone have a good reference for the method of giving a topology to a distributive lattice as outlined in M.H. Stone's "Topological representation of distributive lattices and Brouwerian logics"? ...

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502 views

### Name for a topological space where every closed set contains a closed point

A coauthor and I have stumbled upon a useful topological property -- namely, we are interested in the property that every nonempty closed set contains a closed point. However, neither of us are ...

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849 views

### Largest Hausdorff quotient

The inclusion of the full subcategory of Hausdorff topological spaces into the category of topological spaces has a left adjoint, which can be proven easily by the Adjoint Functor Theorem (see for ...

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334 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 ...

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1k views

### An ultrafilter is a set of subsets containing exactly one element of each finite partition: reference request

There are probably dozens of ways of defining "ultrafilter". The definition I've seen most often involves first defining "filter", then declaring an ultrafilter to be a maximal filter.
But there's ...

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**1**answer

339 views

### Is it possible to define a closure operator in terms of partial ordering?

For boolean algebra, let's take Roman Sikorski's Boolean Algebras as our reference. After giving a set of axioms, he proves (p.9) that the join of A and B is the least element of the algebra such that ...

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696 views

### Ring of continuous functions, reference request.

I am looking for a reference for the following facts in functional analysis and topology. (If these "facts" are not true, I suppose I'm looking for the closest approximation which is true.)
Let $X$ ...

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340 views

### A question concerning separate and joint continuity of bilinear maps

Suppose that $V$ is a locally convex topological vector space and $f:V^2 \to V$ is a bilinear map. Suppose that $C \subseteq V$ is compact and convex, $f$ maps $C^2$ into $C$ and
$f \restriction C^2$ ...

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223 views

### What is the origin of the metrization problem for compact convex sets?

The following is an ``old question in analysis:''
Is it true that every perfectly normal compact convex subset of a locally convex topological vector space is metrizable?
Here perfectly normal means ...

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**1**answer

662 views

### Idempotent measures on the free binary system?

Let $(S,*)$ be the free (non associative) binary system on one generator (so $S$ is just the set of terms in $*$ and $1$). There is an extension of $*$ to the space $P(S)$ of finitely additive ...

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**1**answer

921 views

### complement of a totally disconnected closed set in the plane

While preparing a course in complex analysis, I stumbled over a remark in Dudziak's book on removable sets, namely that any totally disconnected $K \subset\subset {\mathbb C}$ must have a connected ...

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902 views

### Is there a “disjoint union” sigma algebra?

I'm looking for a measure-theoretic analogue to the disjoint union topology, or for work on the $\sigma$-algebra generated by canonical injections. More formally:
For an indexed family of sets ...

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2k views

### Why are the integers with the cofinite topology not path-connected?

An apparently elementary question that bugs me for quite some time:
(1) Why are the integers with the cofinite topology not path-connected?
Recall that the open sets in the cofinite topology on ...

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4k views

### Is the boundary $\partial S$ analogous to a derivative?

Without prethought, I mentioned in class once that the reason the symbol $\partial$
is used to represent the boundary operator in topology is
that its behavior is akin to a derivative.
But after ...

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1k views

### Is the space of continuous functions from a compact metric space into a Polish space Polish?

Let $K$ be a compact metric space, and $(E,d_E)$ a complete separable metric space.
Define $C:=C(K,E)$ to be the continuous functions from $K$ to $E$ equipped with
the metric $d(f,g)=\sup_{k\in K}\ ...

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437 views

### Homotopy theory for spanning trees of a graph

I am studying a paper of L. Lovász, ``A homology theory for spanning trees of a graph,'' but professor Babai has told me that Lovász later realized that this work is better framed in the language of ...

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588 views

### Every real-valued continuous function on a closed set of compact Hausdorff space has an extension.

I've noted, that the following fact can be proven in a few lines using $C^*$-algebra theory. I wonder if it has a simple elementary proof or not. Probably you can give me a reference.
Suppose $X$ ...

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331 views

### Is there a list of all connected T_0-spaces with 5 points?

Is there some place (on the internet or elsewhere) where I can find the number and preferably a list of all (isomorphism classes of) finite connected $T_0$-spaces with, say, 5 points?
In know that a ...

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776 views

### End point compactification for metric spaces

Freundenthal introduced ends of topological spaces and the end point compactification of locally compact topological spaces adding one point for each end of the topological space (see here).
For ...

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2k views

### totally disconnected and zero-dimensional spaces

When do the notions of totally disconnected space and zero-dimensional space coincide? From what I gather, there are at least three common notions of topological dimension: covering dimension, small ...

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531 views

### Expressing any f(x,y) using only addition and unary functions?

Suppose we have a continuous function $f:R^2\rightarrow R$. I was told of the following remarkable theorem: $f$ can be expressed as the composition of continuous unary functions (that is, functions ...