Skip to main content

All Questions

1,339 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
5 votes
0 answers
107 views

Heuristics for the word problem for monoids

The question is about a purely practical problem: Given is a list of identities, as in http://www.findstat.org/MapsDatabase/Mp00069: ...
Martin Rubey's user avatar
  • 5,822
5 votes
0 answers
144 views

Do products of distance functions separate points?

Let $(X,d)$ be a metric space without isolated points and of diameter $1$. Let $Y=\{y_m\}_{m=1}^{\infty}$ be a dense subset of $X$. Define $g_0\equiv 1$, and for $m>0$ let $g_m=d(\cdot,y_1)\dotsm d(...
erz's user avatar
  • 5,529
5 votes
0 answers
182 views

Which metrizable spaces can be embedded into the countable power of $\omega$ with the cofinite topology?

Let $\omega_{cf}$ be the countable space $\omega=\{0,1,2,3,\dots\}$ endowed with the cofinite topology $$\tau_{cf}=\{\emptyset\}\cup\{U\subseteq\omega:\omega\setminus U\mbox{ is finite}\}.$$ It is ...
Taras Banakh's user avatar
5 votes
0 answers
364 views

Computing the infinite dimensional Lebesgue measure of "cubes"

There is no Lebesgue measure in infinite dimensions—this slogan is familiar to every student interested in analysis. One possible, precise statement of this result may be as follows: if $X$ is an ...
truebaran's user avatar
  • 9,340
5 votes
0 answers
265 views

Uncountable subspaces of the real line

Definition 1. A topological space $\langle X, \tau \rangle$ is meager if $X = \bigcup_{n \in \omega}A_n$, where each $A_n$ is nowhere dense in $\langle X, \tau \rangle$. Definition 2. A topological ...
Smolin Vlad's user avatar
5 votes
0 answers
129 views

Is there an orbit map without path lifting property?

I am looking for an example of a topological group $G$ acting by homeomorphisms on a metrizable space $X$ such that the orbit map $X\to X/G$ doesn't have the path lifting property, that is, there is a ...
Igor Belegradek's user avatar
5 votes
0 answers
132 views

Infinite connected $T_2$-space with fixed-cardinality fibers

What is an example of a connected $T_2$-space $(X,\tau)$ with $X$ infinite and the following property? If $\alpha \leq |X|$ is a non-empty cardinal, then there is a continuous map $f:X\to X$ such ...
Dominic van der Zypen's user avatar
5 votes
0 answers
481 views

Open convex hull of a closed set

Let $X$ be a closed set in a Euclidean space of finite dimension and suppose that its convex hull $H$ is open. I can prove that, in this case, $H$ is a Cartesian product of a line with an open convex ...
David Eppstein's user avatar
5 votes
0 answers
113 views

Free topological groups as smooth infinite-dimensional manifolds

It is known that the Graev free topological group $F(I)$ of the segment $I=[0,1]$ is homeomorphic to $\mathbb R^\infty=\lim\limits_{\longrightarrow}\mathbb R^n$. Is it possible to make an ...
Lviv Scottish Book's user avatar
5 votes
0 answers
316 views

Polish groups with no small subgroups

Definitions. A Polish group is a topological group $G$ that is homeomorphic to a separable complete metric space. A group $G$ has no small subgroups if there exists a neighborhood $U$ of the identity ...
Jackson Morrow's user avatar
5 votes
0 answers
237 views

Polish transversals

A subset of $X$ an indecomposable continuum $Y$ is called a composant transversal if $X$ has exactly one point from each composant of $Y$. So a continuum has a composant transversal precisely when ...
D.S. Lipham's user avatar
  • 3,317
5 votes
0 answers
472 views

Partitioning $\mathbb{R}^n$ into closed sets

Let $n$ be a positive integer. It is well-known that $\mathbb{R}^n$ cannot be non-trivially partitioned into open sets, since it is connected. Let $\frak P$ be a partition of $\mathbb{R}^n$ into ...
Dominic van der Zypen's user avatar
5 votes
0 answers
207 views

Is unitary group paracompact?

In this paper Martin Schottenloher notices that the unitary group $U(H)$ of a separable Hilbert space $H$ is metrizable in the strong operator topology. As a corollary (see R.Engelking, 5.1.3), it is ...
Sergei Akbarov's user avatar
5 votes
0 answers
112 views

Is there a homogeneous compactum where non-empty $G_\delta$s have non-empty interior?

A space $X$ is called an almost $P$-space if $Int(G) \neq \emptyset$ for every non-empty $G_\delta$ subset $G \subset X$. Every $P$-space (that is, a space where $G_\delta$s are open) is an almost $P$...
Santi Spadaro's user avatar
5 votes
0 answers
228 views

What is the smallest number of hyperplanes covering $\ell_2$?

For a Banach space $X\ne \{0\}$, let $\mathrm{cov}_H(X)$ be the smallest number of hyperplanes covering $X$. By a hyperplane in a Banach space I understand any closed affine subspace of codimension ...
Taras Banakh's user avatar
5 votes
0 answers
175 views

Continuous open self maps on Cantor space

A continuous self map on the Cantor space $C = \{0,1\}^\mathbb{N}$ is a mapping $f = (f_i)_{i\in \mathbb{N}}$ such that each $f_i$ is a map from $C$ to $\{0,1\}$ that depends only on a finite number ...
alesia's user avatar
  • 2,772
5 votes
0 answers
102 views

Universal and strong $Q$-sets

A subset $X\subset \mathbb R$ is called $\bullet$ a $Q$-set if for every subset $A\subset X$ there exists a $\sigma$-compact set $C\subset\mathbb R$ such that $C\cap X=A$; $\bullet$ a strong $Q$-set ...
Taras Banakh's user avatar
5 votes
0 answers
128 views

Is the Baireness a 3-space property of topological groups

It is known that the product of two Baire spaces can be meager. On the other hand, by a recent result of Li and Zsilinszky the product of two Baire spaces is Baire if one of the spaces is countably ...
Taras Banakh's user avatar
5 votes
0 answers
204 views

homotopy type of box topology.

Suppose that $X$ is weakly equivalent to a point. Let $I$ be a set. Does $\prod_{i\in I}X$ weakly equivalent to a point, where $\prod_{i\in I}X$ is equipped with box topology ?
Ofra's user avatar
  • 1,613
5 votes
0 answers
157 views

For which topological spaces does pullback along $\operatorname{ev}_0:B^I\to B$ have a right adjoint?

Let $B$ be a topological space. Consider the evaluation at zero of paths in $B$. This is a continuous map $\operatorname{ev}_0:B^I\to B$ where the domain carries the compact-open topology. For which ...
Arrow's user avatar
  • 10.5k
5 votes
0 answers
747 views

Questions about obstruction theory (Hatcher's book)

I'm actually studying obstruction theory as presented in the last section of chapter $4$ of the book Algebraic Topology by Allen Hatcher. He first finds condition so that a space $X$ admits a ...
Diego95's user avatar
  • 521
5 votes
0 answers
212 views

An easier example of complete lattice such that the Scott topology on it is not sober

Basic notions: $1$, A partially ordered set is a dcpo if each of its directed subsets has a supremum. (https://en.wikipedia.org/wiki/Complete_partial_order)\ $2$, A subset O of a dcpo P is called ...
Zhenchao Lyu's user avatar
5 votes
0 answers
171 views

(When) is the Dedekind-MacNeille completion of a po-set Hausdorff?

Let $X$ be a p.o. Consider the topology on $X$ generated by $$U_{x}^{-}:=X\setminus (x\uparrow),\quad U_{x}^{+}:=X\setminus (x\downarrow), \quad x\in X$$ Throughout this discussion I shall refer to ...
Thomas's user avatar
  • 263
5 votes
0 answers
330 views

The second dual of $C(X)$ with the compact-open topology

Let $X$ be a compact Hausdorff space. Then $C(X)$ is a Banach algebra with the supremum norm and so is $A=C(X)^{**}$ under either Arens product. Moreover, it is easy to verify that $A\cong C(Z)$ for ...
user124775's user avatar
5 votes
0 answers
349 views

Tietze extension theorem for lower semi continuous functions

On the Tietze extension theorem, if instead of a continuous function "f" we use a lower semi continuous function on a closed subspace of a metric space, is the theorem correct? I mean, can we extend ...
M. Reza. K's user avatar
5 votes
0 answers
214 views

On generically Haar-null sets in the real line

First some definitions. For a Polish space $X$ by $P(X)$ we denote the space of all $\sigma$-additive Borel probability measures on $X$. The space $P(X)$ carries a Polish topology generated by the ...
Taras Banakh's user avatar
5 votes
0 answers
263 views

Are continuous self-maps of the Golomb space $\mathbb G$ dense in the space of all self-maps of $\mathbb G$?

The Golomb space $\mathbb G$ is the set $\mathbb N$ of positive integers endowed with the topology generated by the base consisting of arithmetic sequences $a+b\mathbb N_0:=\{a+bn:n\ge 0\}$ with $a,b$ ...
Taras Banakh's user avatar
5 votes
0 answers
138 views

Disjoint covering number of an ideal

Let $\mathcal I$ be a $\sigma$-ideal with Borel base on an uncountable Polish space $X=\bigcup\mathcal I$. Let $\mathrm{cov}(\mathcal I)$ (resp. $\mathrm{cov}_\sqcup(\mathcal I)$) be the smallest ...
Taras Banakh's user avatar
5 votes
0 answers
339 views

What is the local structure of a fibration?

It's sometimes said that a fibration is a fiber bundle which is not locally trivial. I'd like to make this precise, by identifying the "local models" on which fibrations are modeled. Here I'd like ...
Tim Campion's user avatar
5 votes
0 answers
64 views

Characters on monotone functions

Characters on the semigroup $(C_{+}^{b}(\mathbb{R}^{d}),+)$, i.e. on bounded positive continuous functions with the ususal pointwise addition, are known to be of the form $C_{+}^{b}(\mathbb{R}^{d})\ni ...
Tobsn's user avatar
  • 289
5 votes
0 answers
395 views

Derived tensor products and Tor of commutative monoids

Two commutative monoids $M,N$ have a tensor product $M\otimes N$ satisfying the universal property that there is a tensor-Hom adjunction for any other commutative monoid $L$: $$\text{Hom}(M\otimes N,L)...
John Berman's user avatar
5 votes
0 answers
211 views

A strict directed colimit of Hausdorff locally-convex spaces that is not Hausdorff

We work in the category of locally-convex spaces (morphisms are the continuous linear maps). Let $\Lambda$ be a directed set, for every $\lambda \in \Lambda$ let $V_{\lambda}$ be a locally-convex ...
Jonathan Gleason's user avatar
5 votes
0 answers
233 views

For which topological spaces X does an exponential object Y^X exist for all "nice" topological spaces Y?

A topological space $X$ is exponentiable, meaning that that an exponential object $Y^X$ exists for every topological space $Y$, iff $X$ is core-compact. However, in the only proof I know of that non-...
Alex Mennen's user avatar
  • 2,130
5 votes
0 answers
159 views

Approximating an open bounded set by compact set

Let $U$ be an open bounded subset of $\mathbb{R}^n$, and $K$ a compact subset of $U$. Does there always exist a compact subset $L$ of $U$ that contains $K$, and such that $L$ is a retract of $U$. ...
E.G.'s user avatar
  • 51
5 votes
0 answers
99 views

Zappa-Szép products of the monoid of integers with itself

Question What are all the functions $\alpha , \beta : \mathbb{N} \to \mathbb{N}$ satisfying the following functional equations? $\bullet ~~~~ \alpha(0)=0, \quad \beta(0)=0\\ \bullet ~~~ \...
HeinrichD's user avatar
  • 5,482
5 votes
0 answers
320 views

Unbounded towers and combinatorial cardinal characteristics of the continuum

Update: Perhaps the question is too difficult. I would appreciate, thus, even just comments or related observations. This question assumes familiarity with combinatorial cardinal characteristics of ...
Boaz Tsaban's user avatar
  • 3,104
5 votes
0 answers
637 views

Unique product groups (and semigroups)

A group $G$ is called a u.p.-group (short for unique product group) if for all nonempty finite subsets $A,B\subseteq G$, there exists an element $g\in A \cdot B$ which can be uniquely written as a ...
Pace Nielsen's user avatar
  • 18.7k
5 votes
0 answers
227 views

Filled level sets of harmonic functions

Let $f$ be an enitre function. Define the "filled level set of $f$ as follows: $$A_M(f):=\{z\in{\mathbb C}:\ |f(z)|\le M\}$$ Theorem 1 in Topological Properties of Level-Sets of Entire Functions ...
José A. Prado-Bassas's user avatar
5 votes
0 answers
237 views

On a very weak notion of fibration (of topological spaces)

Suppose that $f:Y \to X$ is a map of topological spaces, and lets assume for simplicity that $X$ is connected. For the fibers of $f$ to compute the homotopy fibers, one would usually want to demand ...
David Carchedi's user avatar
5 votes
0 answers
1k views

Examples of a topological semidirect product

Let $G$ be a compact topological group, and $\operatorname{Aut}(G)$ the group of autohomeomorphisms of $G$. I have proved some (topological) results about the holomorph $G\leftthreetimes \operatorname{...
Ludolila's user avatar
  • 203
5 votes
0 answers
138 views

Is a successor to a successor to the trivial group topology on an Abelian group, totally bounded?

Is there an example of an Abelian group $G$ and group topologies $\cal S$ and $\cal T$ on it such that $\cal S$ is an immediate successor to the trivial topology on $G$ (i.e there is no other group ...
Minimus Heximus's user avatar
5 votes
0 answers
70 views

Does the $D$-property have universal objects?

A space $(X,\tau)$ is called a $D$-space if whenever one is given a neighborhood $N(x)$ of $x$ for each $x\in X$, then there is a closed discrete subset $D\subseteq X$ such that $\{N(x): x\in D\}$ ...
Dominic van der Zypen's user avatar
5 votes
0 answers
219 views

Topological Subset Take-Away

David Gale's subset take-away game is a game where two players A and B play with a finite set $S$. Players alternately choose proper nonempty subsets of $S$ such that if a subset is chosen, then none ...
user avatar
5 votes
0 answers
207 views

Can the compactification of a (co)tangent bundle equipped with Saski metric be viewed as a "Wick rotation"?

We can equip the (co)tangent bundle of a Riemannian manifold (B,g) with a Saski metric $\hat{g}$ (see, for example, "On the geometry of tangent bundles" by Gudmunssun & Kappos) that looks like \...
harry's user avatar
  • 51
5 votes
0 answers
195 views

Inverse limit in shape theory

Is the shape theory of Hausdorff compact spaces complete with respect to the inverse limit operation?--complete means that for every inverse system of Hausdorff compact spaces, and the shape morphisms ...
Włodzimierz Holsztyński's user avatar
5 votes
0 answers
101 views

The name for the quotient property

I asked this question on math@stackoverflow and was suggested to ask it here as well. We call a surjective $f:X\rightarrow Y$ a quotient mapping if it satisfies, for every $U\subset Y$ (continuity,...
Vadim's user avatar
  • 187
5 votes
0 answers
137 views

Pseudovarieties of monoids

All (pseudo)varieties considered here are (pseudo)varieties of monoids. It is known that any (finite or infinite) monoid that satisfies the identities \begin{equation} xhxyty = xhyxty, \quad xhytxy=...
E W H Lee's user avatar
  • 563
5 votes
0 answers
164 views

Group topologies on $\Bbb Z$ with dense open sets in $\Bbb T$

Let $\Bbb Z$ be embedded in the circle group $\Bbb T$ by an irrational rotation and regard $\Bbb Z$ as a subgroup/subspace of the topological group $\Bbb T$. Are there group topologies $\mathcal A$ ...
Minimus Heximus's user avatar
5 votes
0 answers
912 views

Decreasing sequence of closed convex sets in a Banach space

Let $(C_n)$ be a decreasing sequence of closed convex subsets in a Banach space $(E,\Vert \cdot \Vert)$. The question I have is about the content of $C=\bigcap_{n=0}^\infty C_n$. If the $C_n$ are ...
mathcounterexamples.net's user avatar
5 votes
0 answers
265 views

Quotienting disk inside sphere result in sphere

Let $S^k$ be a topological $k$-dimensional sphere. Let $D^k$ be a $k$ dimensional disk that includes in $S^k$. Let $q: D^k \to D^r$ be a map and $r \leq k$. Let $$W = S^k \sqcup D^r/\sim$$ where $S^...
Prasit's user avatar
  • 2,023

1
5 6
7
8 9
27