Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
0
votes
0answers
25 views
Sequence of solutions of approximated problem converging to stationary solution of the original problem
Let $I(x)$ be the indicator function defined as $$I(x) \triangleq \begin{cases} 1, & x \geq 0 \\ 0, & x < 0 \end{cases}$$ and let $L(x,\alpha) \in [0,1]$ be a smoothing function that ...
-6
votes
0answers
42 views
Prove continuity of F [closed]
Let f:R1->R1 be a continuous function. Prove that the mapping F: C[0,1] -> C[0,1], where Fx(t)=f(x(t)), is continuous.
3
votes
1answer
133 views
Non-Polish Lebesgue probability space?
Any Lebesgue probability space is mod. 0 isomorphic to some Polish probability space (with $\sigma$-algebra the completion of Borel algebra, and some Borel probability). I would like to see an example ...
3
votes
0answers
74 views
A question about cardinal functions
In the context of topological groups, one has $w(X)=d(X)\chi(X)$, where $X$ is a topological group and $w$, $d$ and $\chi$ are the weight, the density and the character, respectively. Since $d(X)\leq ...
1
vote
1answer
140 views
Tietze's extension theorem for compact subspaces
The topological question:
Are there Hausdorff topological spaces $X$ which are compactly generated (=Kelly spaces = $k$-spaces, that is, a subset is closed if its intersection with every compact set ...
-1
votes
0answers
22 views
A set is open in two metric spaces? [migrated]
For $(X,d_1)$ and $(X_2,d_2)$ is two metric space, set $(X=X_1 \times X_2)$
and $d(x,y)=max\{d_1(x_1,y_1),d_2(x_2,y_2)\}$ , $\overline{d}=d_1(x_1,y_1)+d_2(x_2,y_2)$ with $x=(x_1,x_2),y=(y_1,y_2)\in ...
6
votes
0answers
124 views
Zariski-homeomorphisms
This question is motivated by two questions at MO and
at MSE.
I am interested in homeomorphism types of (irreducible) complex-projective varieties with respect to the Zariski topology. Any two ...
15
votes
1answer
433 views
Is a left topological group which is a manifold a topological group?
Let $G$ be a left topological group, i.e. a topological space with group operation such that left multiplication $L_g : x \mapsto gx$ is continuous (but right multiplication and inversion are not ...
0
votes
1answer
202 views
Computing the fundamental group of a flag variety
Let $G$ be a compact and connected and simply connected Lie group and $\mathfrak{g}$ be its Lie algebra and $x\in\mathfrak{g}^*$. How can we compute the fundamental group of $G/G_x$ where $G_x$ is ...
-3
votes
2answers
78 views
The boundary of this set is piecewise smooth? [closed]
Consider a sequence of open sets in $R^n$: $\Omega_1 \supset \Omega_2 \supset\cdots$. Consider that this sets are bounded, convex with the boundary piecewise smooth .When i say smooth i mean ...
1
vote
0answers
100 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 ...
2
votes
0answers
64 views
The Haar integral on uniform spaces
Does anybody know what the current research status is on the topic of Haar measures on uniform spaces? I'm specifically interested in compact uniform spaces and its corresponding Haar probability.
As ...
0
votes
1answer
127 views
US does not imply AB
We say that a topological space $X$ is:
$AB$, provided that $X$ is $T_1$ and for each pair $(A, B)$ of compact, disjoint subsets of $X$ there is $U$, an open subset of $X$, such that either $A ...
2
votes
0answers
40 views
A construction with Hyperspace of continums
Let $X$ be a compact connected metric space. Its hyperspace is denoted by $2^{X}.$ $X$ is considered as a subset of $2^{X}$ via the embedding $x\mapsto \{x\}$. Assume that $f:X\to X$ is a ...
2
votes
0answers
135 views
Homeomorphisms between infinite-dimensional Banach spaces and their spheres
As I know Cz. Bessaga has proved that an infinite-dimensional Banach space is homeomorphic to its unit sphere.
Unfortunately I do not have his book but I want to know is this theorem true without ...
5
votes
2answers
189 views
Can Hausdorff dimension make sets into a Tropical Semiring?
If $X$ is a metric space, we construct Hausdorff $d$ measure from the outer measure
\begin{equation}
H^d(U) = \lim_{\delta \to 0}\inf\left\{\sum_{i=1}^\infty \left(\text{diam}(E_i)\right)^d : ...
0
votes
0answers
59 views
Problem on infinite dimensional metric space, with rigidity assumption
By inspiring from this answer of S. Ivanov, here is a specialization with a rigidity assumption.
Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces verifying :
...
-1
votes
0answers
16 views
Proving a topological space is/is not Hausdorff? [migrated]
I know this is a basic question, but I am having trouble proving that particular topological spaces are/are not Hausdorff and was wondering if I could get some guidance. For example, I have to decide ...
2
votes
1answer
94 views
A question about small cardinals related to Michael's Problem
I'm studying some applications of small cardinals related to the Michael's Problem. Recall that we say that a space $X$ is a Michael space if X is a regular Lindelöf space such that $X\times ...
0
votes
0answers
64 views
What do sparse sets in a norm topology look like in the weak* topology?
I'm wondering if a very "sparse" set in a normed vector space can look connected in the weak* topology. Specifically,
Let V be a Banach space, V* its dual, and X a (uncountable) subset of the unit ...
0
votes
0answers
28 views
hyperspaces and selection principals
Two things bother me for which I haven't found an answer yet:
1.Is anyone familiar with an example of a topological space $X$, in which the hyperspace $2^X$ with the upper Fell topology is ...
1
vote
0answers
46 views
A categorical analogue of Debreu's independent factors theorem
Background
A major question in Decision Theory is that of the cardinal meaning of a utility function. That is, given a set $X$, a utility function $u:X\rightarrow \mathbb{R}$ represents the choices ...
3
votes
1answer
181 views
Density of linear functionals in $L^2$
Let $X$ be a locally convex topological linear space, and let $\mathbb P$ be a probability measure on $X$. Suppose that $\operatorname{var}(\varphi) < \infty$ for all continuous linear functionals ...
1
vote
2answers
190 views
Existence of non-locally constant functions
Given a nondiscrete compact Hausdorff space $K$, does there always exist a real-valued function $f$ on $K$ that is not locally constant? Why/why not?
In http://arxiv.org/abs/math/9505204 the authors ...
0
votes
0answers
44 views
subspace in pseudotopological space
Every topological space gives rise to a pseudotopological space. Conversely, if $X$ is a pseudotopological space then we can define a topology on $X$ such that every filter converging to $x$ in the ...
2
votes
0answers
118 views
Convex hulls of compact sets
Let $A$ be a compact set in a separable Hilbert space $H$, and let $\bar A$ denote its convex hull. Is $\bar A$ compact?
3
votes
1answer
204 views
What is the Stone–Čech compactification of a dense set of $\beta N \setminus N$?
Is the Stone–Čech compactification of a dense $G_\delta$-set $X \subset\beta N \setminus N$ homeomorphic to $\beta N \setminus N$? Here, $\beta N \setminus N$ is the complement of $N$ in the ...
3
votes
2answers
316 views
The quotient of $\mathbb{R}^{n}$ by a closed subset
Let $A$ be a closed subset of $\mathbb{R}^{n}$. Can the quotient space $\mathbb{R}^{n}/A$ be embedded in some Euclidean space $\mathbb R^{m}$? In particular, assume that $A$ is an algebraic variety of ...
16
votes
1answer
446 views
Is Grothendieck classification of tensor norms and Kuratowski's 14 sets theorem somehow related?
It is known that there are only 14 reasonable tensor norms in $Ban$. On the other hand it is well known fact for topologists that one can obtain only 14 different sets from a given set applying ...
6
votes
2answers
146 views
Fixed points and their continuity (2)
Yesterday I asked a question about fixed point. Here is the link.
In summary, the question was,
Let $f : I^2 \to I$ be a continuous map, where $I := [0,1]$ is the unit interval. It is a basic ...
6
votes
1answer
151 views
Fixed points and their continuity
Let $f : I^2 \to I$ be a continuous map, where $I := [0,1]$ is the unit interval. It is a basic fact that for each $y\in I$, the function $x \mapsto f(x,y)$ admits a fixed point. I want to ask whether ...
10
votes
2answers
359 views
subsets of groups which have to be closed no matter what
One example of a subset of a group $G$ which has to be closed in any topology on $G$ compatible with the group operations is a centraliser. Are there any other interesting examples?
-1
votes
1answer
97 views
Is the countably infinite product of locally convex topological vector spaces locally convex?
Let $(X,\tau)$ be a locally convex topological vector space and denote the product space
$$X^{\infty}=X\times X\times X\cdots:=\big\{x=(x_i)_{i\geq 1}:~ x_i\in X\big\}$$
If we endow $X^{\infty}$ ...
3
votes
1answer
131 views
Constructible subset of constructible set
Let $X$ be a topological space. Let $F \subset E \subset X$ be subsets. Assume that $E$ is constructible in $X$ and that $F$ is constructible in $E$. Is it true that $F$ is constructible in $X$?
We ...
9
votes
3answers
302 views
Is every T0 2nd countable space the quotient of a separable metric space?
Suppose the space $X$ has a countable basis and $X$ is $T_{0}$. Must there exist a separable metrizable space $Y$ and a quotient map q:$Y \rightarrow X$?
(Some surrounding facts:
Every metrizable ...
1
vote
0answers
61 views
equivalence of topologies defined on $M_1$(a subspace of bounded measures on $\mathbb{R}$)
Let $\mathcal{C}:=\mathcal{C}(\mathbb{R})$ be the space of continuous functions on $\mathbb{R}$ and $\mathcal{C}_b$ its subspace consisting of bounded elements. Define for $\phi(x):=1+|x|$,
$$
...
2
votes
0answers
90 views
Hall's paper on the profinite groups and Andre Weils “voisinage” notion
I am reading through a classical paper A Topology for Free Groups and Related Groups
by Marshall Hall Jr. in which profinite groups are defined for the first time.
There he defines on p. 129:
...
3
votes
0answers
146 views
Strong measure zero sets and selection principles
A set of reals $X$ is strong measure zero if for any sequence of positive real numbers $ ( \epsilon_n ) _{n \in \omega }$ there is a sequence of open intervals $ ( a_n ) _{n \in \omega }$ which ...
1
vote
1answer
152 views
Interior of a dual cone
Let $K$ be a closed convex cone in $\mathbb{R}^n$. Its dual cone (which is also closed and convex) is defined by $K' = \{ \phi\ |\ \phi(x) \geq 0,\ \ \forall x \in K\}$.
I know that the interior of ...
12
votes
2answers
265 views
compact-open topology on $B(H)$
In topology, it is common to use the compact-open topology on the set of continuous maps between two given topological spaces.
Let now $H$ be a Hilbert space and $B(H)$ the set of continuous linear ...
3
votes
1answer
299 views
Is there a simple topological proof for a topological theorem about $S^2$?
Consider the problem of coloring each point of $S^2$ with one of two colors (say "black" or "white") so that among any three points of $S^2$ which are the vertices of an equilateral spherical triangle ...
9
votes
1answer
185 views
Is it always possible to “encircle” exactly $n$ points in an infinite subset of $\mathbb{R}^d$ without limit points?
Let $d$ be a positive integer, and let $\mathbb{R}^d$ be endowed with the
Euclidean metric. Given an infinite set $S \subset \mathbb{R}^d$ without limit points
and a positive integer $n$, is there ...
2
votes
1answer
91 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 ...
9
votes
0answers
144 views
Multiplicity of ball covering
Background. My questions are motivated by the following:
A. Conway and Sloane in "On the covering multiplicity of lattices" (Discrete
and Computational Geometry, 8 (1992) 109-130) considered the ...
8
votes
2answers
303 views
Totally disconnected locally compact Hausdorff spaces
Can any totally disconnected locally compact Hausdorff space be written as a disjoint union of subsets that are both compact and open?
If this is true, does anyone know of a good reference?
1
vote
0answers
74 views
Topologies on spaces of linear sections
Let $X$ and $Y$ be topological linear spaces which are complete & Hausdorff, and admit dual spaces which separate points. Suppose the topologies are non-separable and non-metrizable.
Let $f : X ...
12
votes
7answers
633 views
Examples of toposes for analysts
I've read that toposes are extremely important in modern mathematics, but I find the definitions and examples given on the nLab page a little too abstract to understand.
Can you provide some examples ...
2
votes
1answer
125 views
characterization of the unit disk
We know of a charcterization of spaces homeomorphic to [0,1], as being metric continua with 2 noncut points. We have as well a characterization of spaces homeomorphic to the unit circle. I can't find ...
5
votes
1answer
118 views
Constructing a odd homeomorphism between $A$ and $S^n$
I have asked this question here seven months ago and until now I got no answer.
Let $A\subset\mathbb{R}^N\setminus\{0\}$ be a closed symmetric set ($x\in A$ then $-x\in A$). Suppose that $A$ is ...
2
votes
1answer
101 views
Tensor product of topological abelian groups with the reals
Given an abelian group A, the tensor product $A \otimes R$, where R are the reals, is naturally an R-vector space.
Now suppose that A is a topological abelian group (if necessary, we can assume it ...
