All Questions
5,185 questions
1
vote
1
answer
129
views
extending disjoint open subsets of a normal Hausdorff space
Under what assumptions on $C$ and $X$ is the following true ? I was neither able to find a counterxample or prove this, though it appears that compactness, e.g. assuming $X$ is compactly generated, ...
- 81
1
vote
1
answer
267
views
Limit points and Homeomorphism
I was asking this question at Mathematics SE but I got nothing at all. This is why I am trying this site.
We consider the topology of the extended real line. Let $h\colon [-\infty,\infty]\to\Bbb R$ ...
- 179
1
vote
1
answer
155
views
Do Locally Contractible, Path-Connected Groups have Accessible Bases?
Suppose $G$ is a locally contractible, metric, path-connected topological group. In my particular case, $G$ will be the group of orientation-preserving homeomorphisms of the plane, denoted $Aut(\...
- 439
1
vote
1
answer
259
views
Is the topology generated by the convergence of finite-dimensional distributions metrizable?
Let $\mathbf{D} := D([0,1]; \mathbb{R}^d)$ be the Skorokhod space (equipped with the Skorokhod metric) of càdlàg functions, and let $X = (X_t)_{t \geq 0}$ be its canonical process. The space of ...
- 309
1
vote
1
answer
225
views
Fixed points of one-point-compactification
Let $M$ be a locally compact (Hausdorff) space, and $g:M\to M$ an isomorphism (think of an action of a finite cyclic group).
By some generalities one can show that the "obvious" map $(M^g)^+\...
- 151
1
vote
1
answer
82
views
Isotopy Classes of Non-Connected Planar Sets
I was just looking through some of my old questions on StackExchange and noticed that this one went totally unresolved. I still think it'd be useful as a lemma for plane topology if it were true, and ...
- 439
1
vote
1
answer
281
views
Separating compact sets in locally compact spaces
It is known that not every locally compact Haussdorff space is normal, see for example
here
But it seems that the following is true, I just want to make sure I am not making any mistake:
Lemma Let ...
- 2,071
1
vote
1
answer
234
views
Density and the projective tensor product
Let $X$ be a locally convex space (over $\mathbb{R}$), $D\subset X$ be dense, $B$ be a Banach space (again over $\mathbb{R}$) with Schauder basis $\{b_i\}_{i =1}^{\infty}$. Is the set
$$
D^+\...
- 5,405
1
vote
2
answers
267
views
The "higher topology" of countable Scott sets
Fix some computable bijection $b$ between $\omega$ and $2^{<\omega}$. For $r\in 2^\omega$, let $$[r]=\{f\in 2^\omega: \forall\sigma\prec f(b^{-1}(\sigma)\in r)\}$$ be the closed subset of Cantor ...
- 20.5k
1
vote
1
answer
102
views
Metrization of quotient spaces defined by sequences of continuous functions
Let $K$ be a compact Hausdorff space and let $C(K)$ be the space of all scalar-valued continuous functions on $K$. Let $(f_{n})_{n}$ be a sequence in $C(K)$ satisfying $\sup\limits_{n}\sup\limits_{t\...
- 3,343
1
vote
1
answer
77
views
What is the minimal possible size of a subset of this semigroup satisfying the following conditions?
Suppose $A$ is some set. Let's define a pair semigroup over $A$ as $P[A] = (A\times A \cup \{0\}, \circ)$, where the $\circ$ operation is defined by the following two identities:
$\forall a \in P[A]$ ...
- 2,618
1
vote
1
answer
163
views
Internal commutative monoid gives commutative monad
Let $(C,\otimes,1)$ be a symmetric monoidal category. Let $(M,\mu,\eta)$ be an internal commutative monoid object.
The functor $X\mapsto M\otimes X$ has a canonical monad structure, with unit and ...
- 2,129
1
vote
1
answer
124
views
Is there some characterization of $\omega^\omega$-base related to $S_\omega$?
For a topological space $X$ and one point $x\in X$, we call the cofinal type of neighborhood bases of $x$ are cofinally finer than $\omega^\omega$-base if for any neighborhood base $\mathfrak{N}$ of $...
- 123
1
vote
1
answer
150
views
A Uniform Metric Selection Theorem
Let $X$ and $Y$ be bounded complete separable metric spaces. Let $C = 2^\omega$ be Cantor space with its standard metric. All product spaces are taken to have the max metric.
Let $F, G \subseteq X\...
- 13k
1
vote
1
answer
453
views
Topology of length spaces
How wild can the topology of a length space be? That is,
Questions:
Let $X$ be a metric space where the distance between two points $x,y \in X$ is the infinum of lengths of rectifiable paths from $...
- 64k
1
vote
1
answer
87
views
Hausdorff quotient collapsing and separating a prescribed collection of disjoint closed subsets
Let $X$ be a compact Hausdorff space (I don't mind assuming it's metrizable).
Let $A_i$ $i\in \mathbb{N}$ be a collection of disjoint closed subsets of $X$.
My question: Does there exist a Hausdorff ...
- 2,964
1
vote
1
answer
177
views
Is the infinite product map $(∏_{i=1}^{∞}S_{i})×f$ topologically transitive
Let $f:X→X$ be a map. We say that $f$ is topologically mixing if for every open subsets $U,V$ of $X$, there exists $N$ such that for every $n≥N$ the set $f^{n}(U)∩V$ is non-empty.
Let $S : X → X$ and ...
- 1,197
1
vote
1
answer
173
views
Does exponential law bijective implies evaluation map continuous?
Husemoller in his "Fibre Bundles" writes that the exponential law $$ \theta \colon B^{A \times X} \to (B^X)^A $$ which is always injective, is bijective if and only if the evaluation map
$$ Ev \colon ...
- 1,346
1
vote
1
answer
122
views
Irreducible subcontinuum of Lorenz attractor?
In my first question Lorenz attractor path-connected?, some are saying the Lorenz attractor $\mathscr L$ is not path-connected.
But suppose $x$ and $y$ are two points in different path components of ...
- 331
1
vote
1
answer
399
views
Maximum of a sum of Gaussian functions
Consider the function which maps $\mathbb{R}^n$ to $\mathbb{R}$
\begin{align}
f(x) = \sum_{i=1}^{n} b_i\phi_i(x)
\end{align}
where $\phi_i(x) = \exp(-\frac{||x-x_i||_2^2}{2})$ are Gaussian functions ...
- 1,421
1
vote
1
answer
140
views
Slightly finer topology vs a quasi-component
Let $(X,\tau)$ be a topological space, and let $Q$ be a quasi-component of $X$. Let $S$ be a subset of $X\setminus Q$. Then is $Q$ necessarily a quasi-component of $X$ in the topology generated by $\...
- 107
1
vote
1
answer
149
views
Descending almost-contained subsets of $\omega$ [closed]
Let $A$ be an infinite subset of $\omega$ such that $\omega\setminus A$ is also infinite.
Under the Continuum Hypothesis is there a sequence $(A_\alpha)_{\alpha<\omega_1}$ of subsets of $\omega$ ...
- 955
1
vote
1
answer
126
views
Do these sorts of submonoids go by a particular name?
Given any monoid $M$ for every element $x\in M$ we can define two submonoids of $M$ as follows:
$$r(x)=\{y\in M:xy=x\}$$
$$l(x)=\{y\in M:yx=x\}$$
Do these sorts of sub-monoids go by a particular name?...
- 111
1
vote
1
answer
73
views
Why is a certain space of linear isometries paracompact
Let $V$ be a finite dimensional real inner product space and $U$ a real inner product space of countable dimension. Why is the space of linear isometries from $V$ to $U$ paracompact?
- 765
1
vote
1
answer
141
views
Continuum image of line is chainable?
Let $X$ be a metric continuum (compact + connected) which is the one-to-one continuous image of the interval $[0,\infty)$. Such an $X$ is called a linear continuum.
It seems like $X$ should be ...
- 955
1
vote
1
answer
134
views
amalgamated sum of monoids
Consider the amalgamated sum $Q_1 \rightarrow^{v_1} Q_1 \oplus_P Q_2 \leftarrow^{v_2} Q_2$ of $Q_1 \leftarrow^{u_1} P \rightarrow^{u_2} Q_2$ with $Q_1,Q_2,P$ being monoids.
Why does $v:= v_i \circ u_i$...
- 65
1
vote
1
answer
96
views
If $H$ is commutative and unit-cancellative, then so is the monoid of non-empty ideals of $H$
Let $H$ be a (multiplicatively written) commutative monoid with identity $1_H$. Given $X, Y \subseteq H$, we take
$$XY := \{xy: x \in X,\, y \in Y\}.$$
We call a set $I \subseteq H$ an ideal of $H$ ...
- 10.5k
1
vote
1
answer
1k
views
Closure of the interior of a convex set in a topological vector space
Let $C$ be a convex set with nonempty interior in a topological vector space. Do we always have :
$\overline {C^\circ} = \overline{C}$ ?
If not, what is the "minimal" condition on the space so that ...
- 1,035
1
vote
1
answer
375
views
Inverse map of chaotic map : confusion and request for information
This is based from the paper titled, "Chaos-Based Simultaneous Compression and Encryption for Hadoop" in Section 2.3.1 download link
The Authors say that given a symbolic sequence, it can be encoded ...
- 135
1
vote
1
answer
346
views
Convergent sequences in compact spaces
Problem. Assume that a compact space $X$ can be written as the union $X=K\cup D$ of a compact metrizable subspace $K$ and a discrete subspace $D$.
Does $D$ contain a non-trivial convergent sequence in ...
- 42k
1
vote
1
answer
176
views
Interval topology on complete Boolean algebras
Let $(P,\leq)$ be a poset. The interval topology $\tau_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\...
- 48.9k
1
vote
1
answer
136
views
Does any locally compact space have a proper diagonal neighborhood?
Let $X$ be a locally compact Hausdorff space. Does the diagonal $\Delta X \subset X \times X$ have a (closed) neighborhood $N$, such that the canonical projection maps $N \to X$ are proper?
- 4,010
1
vote
1
answer
933
views
Every topological manifold is a ENR? (Reference)
It seems to be widely known that every topological manifold can be embedded as a neighbourhood retract in euclidean space, I can not find a reference, though.
The reason, why I'm asking this, is that ...
- 1,082
1
vote
1
answer
120
views
Lower neighbors in the lattice of topologies
Given any poset $(P,\leq)$ and $x, y\in P$ we set $[x,y) = \{p\in P: x\leq p < y\}$, and $(x,y]$ is defined in an analogous manner. For any set $X$, let $\text{Top}(X)$ denote the set of topologies ...
- 48.9k
1
vote
1
answer
139
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 ...
- 1,112
1
vote
1
answer
437
views
Does order-preserving equal continuous? [closed]
Let $P,Q$ be posets and endow them with the interval topology $\tau_i(P)$ and $\tau_i(Q)$ respectively. Is it true that if $f: P\to Q$ is order-preserving, then it is continuous, and vice versa?
user62017
1
vote
1
answer
149
views
Problem about the existence of a continuous surjective map [closed]
Let $F$ be a closed set in $\Bbb R^2$, $F\neq \varnothing,\Bbb R^2$, and $F^\circ\neq \varnothing$,
does there exist a continuous surjective map from $\Bbb R\times \Bbb Z$ to $F^{\circ-}-F^\circ$？
1
vote
1
answer
359
views
Bounded-open topology vs norm on $L\left(X,Y\right)$
In general topology there is two ways of introducing a topology on the space of (continuous) maps between, say, metric spaces: set-open topology and uniform topology (it is a uniformity of uniform ...
- 5,529
1
vote
1
answer
121
views
A Hausdorff atom in lattice of group topologies
Do you have an example of an infinite Hausdorff nonabelian topological group $(G,\mathcal T)$ such that for any nontrivial group topology $\mathcal S$ on $G$ with $\mathcal S\subseteq \mathcal T$ we ...
- 1,145
1
vote
1
answer
330
views
Two questions on path connected spaces
Is it true to say that a compact hausdorff space $X$ is path connected if and only if for every continuous function $f:X\to \mathbb{C}$, we have $f(X)\subset \mathbb{C}$ is path connected?
2....
- 366
1
vote
1
answer
100
views
Name for (function, set) pairs?
Right now I'm working on a topological graph theory problem. To prove a theorem I introduced some objects. Has anyone heard of something similar before? I would like to call them by the right name.
...
- 1,329
1
vote
1
answer
189
views
constructible set and fibre product
Let $P$ be certain property. Let $S \subset \mathbb{C}^n\times \mathbb{C}^m$ be a set of closed points such that for any point in $S$, it satisfies the property $P$. I know for any $x \in \mathbb{C}^n$...
- 3,472
1
vote
1
answer
118
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 Gelfand--...
- 1,453
1
vote
1
answer
182
views
A space with countable tightness which is not a Fréchet space?
I need a space with countable tightness which is not a Fréchet space. In this space, I am searching for a point with no deleted neighborhood consisting entirely of P-points.
(A P-point is a point $x \...
- 113
1
vote
1
answer
698
views
Bases of completely regular (Tychonoff) spaces
If the space $X$ is completely regular, we know that the collection
{${\rm int}\,Z(f)$:$f$ is a continuous function from $X$ to the real numbers} is an open base for open subsets of the space $X$ (i....
- 1,788
1
vote
3
answers
884
views
Does the manifold of the three dimensional group of rotations SO(3) cause a separation of space in the group of rigid motions SE(3)?
The group of three dimensional rotations $SO(3)$ is a subgroup of the Special Euclidean Group $SE(3) = \mathbb{R}^3 \rtimes SO(3)$. The manifold of $SO(3)$ is the three dimensional real projective ...
- 13
1
vote
1
answer
260
views
The intersection of Block Groups and R-trivial (finite) monoids
Let $\textbf{BG}$ be the pseudovariety of block groups, also known as $\textbf{EJ}, \textbf{PG},\ldots,\text{etc.}$(see [1]), and let $\textbf{R}$ be the pseudovariety of R-trivial monoids, by the ...
- 424
1
vote
1
answer
324
views
Sufficient conditions for Hausdorffness
Let $(X,\tau)$ be a $T_1$ topological space and $Y\subset X$ a dense subspace which is completely metrizable. Are there any sufficient conditions to ensure that $(X,\tau)$ is Hausdorff using the known ...
- 159
1
vote
1
answer
1k
views
Representations of regular maps (four color theorem)
For the scope of the four color problem and without lack of generality, maps can be represented in different ways. This is generally done to have a different perspective on the problem.
For example, ...
- 351
1
vote
1
answer
208
views
When do maps of ineffective orbifolds descend to their effective part?
If $$f:\mathscr{X} \to \mathscr{Y}$$ is a map between (possibly ineffective) orbifolds (in the sense of differentiable stacks, or orbifold groupoids), does it follow that $f$ induces a map between ...
- 15.5k