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

learn more… | top users | synonyms (2)

15
votes
1answer
364 views

A problem of Keisler and Tarski

The following question dates back to Keisler and Tarski: From accessible to inaccessible cardinals, Fund. Math. 53, 1964 and also perhaps Mazur: On continuous mappings of Cartesian products, Fund. ...
3
votes
0answers
81 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 ...
3
votes
0answers
59 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 ...
9
votes
3answers
460 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 ...
5
votes
1answer
111 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 ...
18
votes
2answers
982 views

Is every compact topological ring a profinite ring?

There are a lot of compact (Hausdorff) groups, whereas every compact field is finite. What about rings? Is there a classification theorem for compact rings? If you take a cofiltered limit of finite ...
2
votes
1answer
94 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 ...
4
votes
1answer
80 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 $n\ge0$, $r_n\in R$ ...
0
votes
0answers
36 views

Is preimage of closure equal to closure of preimage under continuous topological maps? [closed]

Let $f:X \rightarrow Y$ be a continuous map of topological spaces and $B \subseteq Y$ Is it true that $f^{-1}(\overline{B})=\overline{f^{-1}(B)}$?
-4
votes
0answers
45 views

Is there exist homeomorphism between (-1,1) x [-1,1) and [-1,1] x [-1,1)? [duplicate]

Is there exist homeomorphism between (-1,1) x [-1,1) and [-1,1] x [-1,1) ?
-4
votes
1answer
143 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
0answers
63 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, ...
13
votes
0answers
303 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) ...
4
votes
3answers
433 views

Characterization of Tychonoff spaces in terms of open sets

Metrizability and complete regularity are topological properties that are, in a sense, different from the Hausdorff condition because they are not defined purely in the terms of the open sets, but ...
2
votes
1answer
63 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 ...
2
votes
1answer
126 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 $S$ ...
2
votes
1answer
314 views

Weaker form of irreducible surjections

An irreducible surjection is usually defined as a continuous closed surjective map $f:X\rightarrow Y$ such that if for some closed set $C\subset X$ one has $f(C)=Y$ then $C=X$. In my dissertation I ...
1
vote
0answers
56 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 ...
2
votes
1answer
83 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 ...
5
votes
1answer
462 views

Spaces over which every vector bundle is a summand of the trivial bundle

Let X be a Hausdorff space such that every real vector bundle on X is summand of a trivial bundle. Does this imply that X is homotopy equivalent to a compact Hausdorf space? This question ...
10
votes
1answer
284 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 ...
3
votes
0answers
52 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 $AB=R$?
6
votes
1answer
211 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 ...
1
vote
0answers
62 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
1answer
52 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 ...
2
votes
2answers
423 views

Is there a countable pseudocharacter Hausdorff space,such that…?

Let X be a Hausdorff space and Difine the Property A as following: if $\mathscr{U}$ is a collection of open sets of X that witnesses Hausdorff property of X (= $\forall x,y \in X$, there exist two ...
11
votes
0answers
175 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 ...
5
votes
1answer
238 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 ...
1
vote
0answers
60 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. ...
6
votes
1answer
321 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 ...
2
votes
1answer
121 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 ...
3
votes
3answers
1k views

Is a connected separable locally euclidean Hausdorff topological space second countable?

This question arose from considering for a connected smooth Hausdorff manifold the (possible) equivalence of the following properties: (1) paracompact, (2) metrizable, (3) second countable, (4) ...
8
votes
1answer
129 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 ...
5
votes
0answers
90 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 ...
3
votes
1answer
177 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
1answer
61 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 ...
3
votes
1answer
128 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
1answer
223 views

A forked plane continuum

I came up with this question while trying to solve the following MO one: Does every connected set that is not a line segment cross some dyadic square? Suppose $C$ is a plane continuum (i.e. a ...
5
votes
3answers
107 views

A Mackey-Ahrens theorem for uniform spaces?

Let $X$ be a uniform space and $F(X)$ the vector space of all uniformly continuous real-valued functions over $X$. It is possible to express every bounded uniform semimetric $d$ on $X$ as $d(x,y) = ...
8
votes
0answers
128 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 ...
2
votes
1answer
72 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 ...
4
votes
1answer
192 views

Is the absolute of a compact space the projective limit of the Stone-Čech compactifications of its open dense subsets?

Is the following statement true, and if it is, does someone have a reference? Let $X$ be a compact (i.e., compact and Hausdorff) topological space. Then the Gleason space (=Iliadis absolute, ...
51
votes
2answers
822 views

Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$

Let $f : \mathbb{R} \longrightarrow \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f(I)$ of $\mathbb{R}^2$. Is it true that there must be a ...
7
votes
1answer
391 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
2answers
328 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. ...
8
votes
2answers
263 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 ...
0
votes
0answers
34 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. ...
3
votes
1answer
128 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 ...
7
votes
1answer
286 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 ...
2
votes
0answers
175 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 ...