All Questions
Tagged with general-topology or gn.general-topology
4,601 questions
0
votes
0
answers
78
views
Pareto-optimal front $F$ in $m$-dimensional space can not have more than $\mathbf{H}_{m-2}(F)$ homology groups
I need to prove that a Pareto-optimal front $F$ in $m$-dimensional space (i.e. $m > 1$) can not have more than $\mathbf{H}_{m-2}(F)$ homology groups.
What it simply means that in a 2-dimensional ...
- 101
0
votes
0
answers
72
views
Li-Yorke sensitivity Vs Li-Yorke dense chaos
Let $X$ be a compact metric space, $X*X$ its cartesian product, and $A$ a subset of $X*X$.
Are the following two properties the same, or e.g. one is stronger than the other?
$A$ is dense and residual ...
- 141
0
votes
0
answers
139
views
Why the name 'regular' space?
It is well known that a regular space is a topological space $X$ with these two properties:
1)All one point sets are closed.
2)For every $x\in X$ and every closed set $B$ (such that $x\notin B$), ...
- 883
0
votes
0
answers
63
views
a lemma on interval translation map
Consider the map $S:T^1 \to T^1$ where $x \mapsto x+c_j$ , mod 1 where $c_j$'s are real numbers. We represent $T^1$ as a union of disjoint subsegments $M_j=[t_j,t_{j+1})$, $j=0 , \cdots ,n , t_0=t_n$....
- 145
0
votes
0
answers
152
views
Left-side cosets of an open subgroup
Let $G$ be a topological group and $H$ its closed subgroup. $K$ and $L$ are open subgroups of $G$ and $H$ respectively. Let $g_{1}, g_{2}\in G$. We assume $L\cap g_{1}K\neq \emptyset$ and $L\cap g_{2}...
- 479
0
votes
0
answers
66
views
A topological space whose connected components are locally connected
We say that $X$ is locally connected at $x$ if for every open set $V$ containing $x$ there exists a connected, open set $U$ with $x\in U\subseteq V$. The space $X$ is said to be locally connected if ...
- 71
0
votes
0
answers
140
views
Can the infinite jungle gym surface be expressed by an exhaustion of compact surfaces with one boundary component?
Is it possible to write the infinite jungle gym surface as the increasing union of compact surfaces whose boundaries consist of only one simple closed curve? I believe that this is true. This surface ...
0
votes
0
answers
371
views
Completness of strong operator topology on norm-bounded sets
Let $H$ be a separable Hilbert space and $B(H)$ the space of bounded linear operators on $H$. It is know, that the strong operator topology is metrizable on norm-bounded sets of $B(H)$. My question is,...
0
votes
0
answers
66
views
Question on existence of almost length-minimizing curve in a general domain?
I have the following question: for a general domain $\Omega$ in $\mathbb{R}^n$, is it true that for each pair of points $x,y\in \Omega$, there exists a curve $\gamma$ connecting $x$ and $y$ in $\Omega$...
- 11
0
votes
0
answers
81
views
Let $S$ be a surface, $K$ compact in $S$ with finitely many components. Does the frontier of a component of $S-K$ have finitely many components?
Let $S$ be a connected surface and $K$ a compact subset of $S$ with finitely many connected components. Let $U$ be a connected component of $S-K$. Does the frontier of $U$ in $S$ have finitely many ...
0
votes
0
answers
217
views
Intersection of zero sets of continuous functions
Let the zero sets $F=\{x \in \mathbb{R}^n: f(x) = 0\}$, $G = \{x \in \mathbb{R}^n : g(x) = 0\}$, where $f$ and $g$ are $m$-dimensional real, analytic, continuous, and nonlinear vector functions. Under ...
- 1
0
votes
0
answers
197
views
Is this topology on $\mathbb{Q}$ well studied?
Let $\|\cdot\|_p$ denote the $p$-adic norm on $\mathbb{Q}$. For the whole set of primes $P$ consider the topology which is generated with prebase of open sets $V_{p,\varepsilon}(x) = \{y\in\mathbb{Q} :...
- 291
0
votes
0
answers
174
views
Problem of Thickening an Arc in a Topological $ 2 $-Manifold
Let $ M $ be a topological $ 2 $-manifold (possibly with boundary), $ C $ an arc in the interior of $ M $ (i.e., an injective continuous function from $ [- 1,1] $ into $ \operatorname{Int}(M) $), and $...
- 1,396
0
votes
0
answers
45
views
Critical Growth of Dimension for Dense Cover by Linear Subspaces
Let $X$ be a separable Banach space of dimensional $>2$. When does there exist a sequence positive integers $\{N_n\}_{n \in \mathbb{n}}$ such that
For any sequence of distinct finite-dimensional ...
- 5,405
0
votes
0
answers
107
views
Open subset of compact-open topology
Let $E$ be a Banach space, $X$ a locally-compact metric space, equip $C(X,E)$ with the compact-open topology. Let $F:E \rightarrow E$ and consider the induced map $F_{\star}(f):=F \circ f$ on $C(X,E)$...
- 5,405
0
votes
0
answers
113
views
Viewing limit as a map
Question: Let $X$ be a set of functions from $\mathbb{R}$ to itself. Consider the subset $X_0$ of the sequences $(f_n)_{n=1}^{\infty}\in X^{\mathbb{N}}$ for which
$$
f_{\infty}(x) = \lim\limits_{n \...
- 5,405
0
votes
0
answers
102
views
Can a quotient space of a locally convex space have finer topology that its domain?
The following is related to this post.
Let $X=X'$, as sets, and let $T:X \rightarrow X'$ be a surjective map from a countably infinite-dimensional LCS $X$ to itself and equip $X'$ with the final ...
- 5,405
0
votes
0
answers
60
views
The scalar convergence in $\mathcal{C}(X)$ is topologizable?
Let $(X,\|.\|)$ be a separable Banach space and $\mathcal{C}(X)$ be the collection of all nonempty, closed and convex subsets of $X$. For any $C$ in $\mathcal{C}(X)$ we set
$$
s(x^*, C) := \sup_{x\in ...
- 115
0
votes
0
answers
84
views
Can we extend a function from the diagonal matrices to an orthogonally-invariant function on $\text{GL}_n$?
This is a cross-post.
Let $g:(0,\infty)^n \to [0,\infty)$ be a symmetric function -i.e. $g(\sigma_1,\dots,\sigma_n)$ does not depend on the order of the $\sigma_i$, with $g(1,\dots,1)=0$.
We ...
- 6,741
0
votes
0
answers
61
views
Weak topology of Gaussian measures
Let us consider a space of Dirac measures $\delta_{x}$ on a Tychonoff space $X$. I know that this space is homeomorphic to $X$. A space of Gaussian measures (weak topology) on some loсally convex ...
- 11
0
votes
0
answers
142
views
Cardinality of the closure of subset of a dense subset
Is the following true? Given a first countable Hausdorff space $X$ and a dense subset $Y\subset X$ which is initially $2^\omega$-initially Lindelöf in $X$. Let $F$ be a subset of $Y$ such that $\vert ...
0
votes
0
answers
53
views
Separability of Minkowski Sum of well-behaved sets
Let $A$ and $C$ be non-empty simply connected and connected subsets of $\mathbb{R}^k$ and suppose that $C$ is convex. Then is the Minkowski sum $A+C$ separable?
- 5,405
0
votes
0
answers
104
views
Locally connectedness and accessibility in $\mathbb{C}$
Suppose $\Omega$ to be a bounded area in the complex plane $\mathbb{C}$ with a locally connected boundary $\partial\Omega$, then every point of $\partial\Omega$ is accessible from Ω.Here accessibility ...
- 101
0
votes
1
answer
100
views
What is the definition of a prorelation?
In the context of quasi-uniform spaces, what is a prorelation?
In the text I'm reading, they're defined as a down-directed upper set on relations X->Y.
Now, I'm fine with a down-directed up-set, but ...
- 1
0
votes
0
answers
66
views
Generalized compact open topology?
Let $X,Y$ be topological spaces. The compact-open topology on $C(X,Y)$ is generated by the sub-basic open sets
$$
\left\{U_{K,O}: \mbox{ K is compact in X and O is open in Y}\right\}\\
U_{K,O}:=\...
- 5,405
0
votes
0
answers
94
views
A characterization for a topological property
Let $X$ be a compact non-Hausdorff topological space. I am looking for a characterization for the following property on $X$:
Property: For each non-empty closed subset $C$ of $X$ there exists a ...
0
votes
0
answers
46
views
Show that $\big(s(. |C_n)\big)_n$ is equicontinuous on $X^*$
Let $(X,\|.\|)$ be a separable Banach space with dual $X^*$. $\mathcal{P}_{wkc}(X)$ be the class of nonempty, weakly-compact and convex subsets of $X$. For any $C\in\mathcal{P}_{wkc}(X)$ we define ...
- 1
0
votes
0
answers
66
views
Coincidence Topologies for $L^p$ spaces
If $X$ and $Y$ are compact metric spaces then it is well-known that the compact-open topology on $C(X,Y)$ coincides with the topology of uniform convergence on compacts. Therefore, the latter is ...
- 5,405
0
votes
0
answers
142
views
Mackey topology
Recall that for a Hausdorff locally convex space $X$ the Mackey topology $\tau (X^*,X)$ is the topology in its topological dual $X^*$ of uniform convergence on all weakly compact absolutely convex ...
- 1
0
votes
0
answers
86
views
Let $f$ be periodic with a continuous image and $a_n = cn$ for some $c > 0$. When is $\{f(a_n)\}$ dense in the image of $f$?
Let $f:\mathbb{R}\to\mathbb{R}$ be a periodic function with period $T$ and continuous everywhere except perhaps on a countable set, and have an image that's an uncountable subset of $\mathbb{R}$. Let $...
- 277
0
votes
0
answers
162
views
A ``1-soft'' improvement of the Parovichenko theorem
This is a ``1-soft'' modification of this problem. We start with the necessary definitions.
Definition 1. A compactification $c\mathbb N$ of the discrete space $\mathbb N$ is called 1-soft if for any ...
- 41.8k
0
votes
0
answers
147
views
Approximation of Inductive Tensor Product $C(X) \bar{\otimes} C(Y)$
The following question is from Banach Algebra Techniques in Operator Theory written by Ronald G. Douglas.
Assume both $X, Y$ are Banach spaces and $X \otimes Y$ is the algebraic tensor product. Let ${...
- 307
0
votes
0
answers
135
views
Are geometric progressions closed in the $p$-adic topology?
For a prime number $p$, the $p$-adic topology on the set $\omega$ of non-negative integers is generated by the base consisting of the arithmetic progressions $x+p^n\omega:=\{x+p^ny:y\in\omega\}$ where ...
- 41.8k
0
votes
1
answer
134
views
A closed subset $B$ of the Hilbert cube such that $\operatorname{Int}(B) = \emptyset$ and $B$ is not a z-set
$\def\cube{\mathbf Q}$Let $\cube$ be the Hilbert cube. Give an example of a closed subset $B$ of $\cube$ such that $\operatorname{Int}(B) = \emptyset$ and $B$ is not a z-set.
If anyone has any idea ...
0
votes
0
answers
68
views
Specific property of borelian sigma-algebras
Let X be a set and S a sigma-algebra on X.
Let us name borelian sigma-algebra on X a sigma-algebra that is generated by a topology T on X. Given that it is possible for a set X that some sigma-...
- 2,655
0
votes
0
answers
117
views
Cardinal Invariants and Physics
There are many applications of topology to physics, but I wonder if there is a known application of cardinal invariants to physics.
0
votes
0
answers
144
views
Making a quasi-compact open into an affine open
Let $X$ be a spectral topological space, $U\subset X$ be a quasi-compact open subspace. Is there necessarily some scheme structure on $X$ (we do not require it to be affine) such that $U$ endowed with ...
user138661
0
votes
0
answers
146
views
A scheme whose underlying space is the product of the underlying spaces of schemes
We know that the product of two spectral topological spaces is spectral.
If $X$ is the underlying space of the scheme $\mathrm{Spec}\,\mathbb{Z}[x]$, what is a simple example of an affine scheme ...
user140269
0
votes
0
answers
325
views
Grothendieck topology on a scheme equivalent to the circle
Suppose a class of morphisms of schemes is reasonable enough so that we can associate a small site to any scheme (just like small étale site). Is there a simple class of morphisms such that there is a ...
user138661
0
votes
0
answers
59
views
A retract algebraic subset of the plane which does not admit an algebraic retraction
What is an example of an algebraic (=Zariski closed) subset $C$ of $\mathbb{R}^2$ which is a topological retract of $\mathbb{R}^2$, but there is no algebraic retraction $P:\mathbb{R}^2 \to C$?
What ...
- 356
0
votes
0
answers
58
views
Does the total space of a bundle satisfy the Tietze extension property when the fiber and base space do satisfy this property?
We say that a Topological space $Y$ satisfies the Tietze extension propery, TE property, if in the formulation of Tietze extension theorem "$\mathbb{R}$" can be replaced by $Y$.
Obvioysly the ...
- 356
0
votes
0
answers
643
views
A new generalization of the dimension?
During my research, I came a cross on these notions :
Definition 1: A structure $S$, is a pair $(X, \mathcal T)$ with $X$ a set and $\mathcal T$ a set of subsets of $X$, stable by arbitrary ...
- 4,074
0
votes
0
answers
63
views
continuous map from $\mathbb R$ to $\mathbb R^2$ that send any convex on a convex [duplicate]
Let $f$ be a continuous fonction from $\mathbb R$ to $\mathbb R^2$, such that for any $a<b\in \mathbb R,\,\, f([a,b])$ is convex.
Is there a line $D\subset \mathbb R^2$ such that $f(\mathbb R)\...
- 469
0
votes
0
answers
93
views
Can we express separability of a ray-remainder in terms of the function algebra?
Let $X = [0, 1)$ be a ray and $C(X)$ the algebra of bounded continuous real functions. The spectrum of $C(X)$ is the Stone-Cech compactification $\beta [0,1) $ of the ray. It's easy to see the ...
- 1,955
0
votes
0
answers
116
views
Open subsets of the n-torus containing no nontrivial loops
Let $T^n$ denote the $n$-dimensional torus. Suppose there is an open subset $U\subset T^n$ not containing any nontrivial loop. Does this imply that the inclusion $U\hookrightarrow T^n$ is ...
user83520
0
votes
0
answers
97
views
Is there any concise sufficient condition for the dual space to have Kadec property?
A normed space $E$ has a
Kadec property if the norm- and weak topologies coincide on the unit sphere of $E$.
Kadec-Klee property if any sequence on the unit sphere, that is weakly convergent is also ...
- 5,529
0
votes
0
answers
84
views
Under what conditions on $\mu^{\beta}$ we have $L_1(\beta X,\mu^{\beta})$ isometrically isomorphic to $L_1(X,\mu)$?
Let $X$ be a locally compact Hausdorff space, $\beta X$ its Stone-Cech compactification and $\Delta: X\to\beta X$ the inclusion map. Given a Borel probability measure $\mu^{\beta}$ over $\beta X$, is ...
- 2,044
0
votes
0
answers
220
views
short exact sequence of profinite groups
Let $A\rightarrow B\rightarrow B/A$ be a short exact sequence of topological groups. Is it true that if there exists a continuous function $B/A\rightarrow B$ (of underlying spaces) such that the ...
- 1,613
0
votes
0
answers
83
views
Topology of sets given by semi-continuous functions
$M$ is a compact space. Assume $f$ is upper semi-continuous on $M$, $g$ is lower semi-continuous on $M$, and $f(x) \geq g(x)$ for any $x\in M$.
If $f(x_0) = g(x_0) $ for some point $x_0\in M$,
Then $...
- 173
0
votes
0
answers
120
views
A topology on the product space of Euclidean space and smooth functions space
I'd like to know if there is a well-known topology on the space $S := \mathbb R \times C^\infty(\mathbb R)$, such that $(x_n, f_n) \to (x, f)$ in $S$ with respect the topology is equivalent to
$$(x_n,...
- 1,399