All Questions
5,185 questions
1
vote
0
answers
192
views
Is the domain space in Lusin's theorem required to be Hausdorff?
I'm reading a general version of Lusin's theorem, i.e.,
If $\mu$ is a finite Radon measure on $X$, and $Y$ is a second countable topological spaces, then for any Borel-measurable function $f:X\to Y$ ...
- 825
1
vote
0
answers
274
views
Functional equation $f(x*y) = f(f(x)*f(y))$
Find all endo-functions $f$ on a commutative semigroup $(\mathbb{S},*)$ such that
$f(x*y) = f(f(x)*f(y))$.
Typical case of interest are $(\mathbb{N},+)$ or $(\mathbb{Z}/k\mathbb{Z},+)$ or $(\mathbb{Z}/...
- 1,306
1
vote
0
answers
181
views
Second homology group of a presentation complex
I am trying to learn results related to the presentation complex of a group and I am new to this subject. So I apologize if the questions are silly.
Given a finite group $G$, and a presentation $P$ of ...
- 179
1
vote
0
answers
100
views
Density of Lipschitz functions in Bochner space with bounded support
Let $X$ and $Y$ be separable and reflexive Banach spaces with Schauder bases. Let $\mu$ be a non-zero finite Borel measure on $X$ and let $L^p(X,Y;\mu)$ denote the (Boehner) space of strongly p-...
- 21
1
vote
0
answers
53
views
The "hyperbolicity preserving" probabilities
A classical fact (due to Polya ?) is that if $P\in{\mathbb R}[X]$ has only real roots (one says that $P$ is hyperbolic), and $a$ is a real number, then the roots of
$$L_aP(X):=\frac12(P(X+ia) +P(X-ia))...
- 52.3k
1
vote
0
answers
81
views
Morphism in commutative square strict?
Let $G,H$ be topological groups and $f:G\rightarrow H$ a continuous group homomorphism.
Then $f$ is said to be strict if $G/\mathrm{Ker}(f) \cong \mathrm{Im}(f)$ is an isomorphism of topological ...
- 473
1
vote
1
answer
106
views
Rothberger property and semi-open sets
Here is the definition of a space $X$ to be Rothberger, if for each sequence $(\mathcal{U}_{n})_{n\in\mathbb{N}}$ of open covers of $X$, there exists a sequence $(U_{n})_{n\in\mathbb{N}}$ where $U_{n}\...
- 13
1
vote
0
answers
48
views
Reference for preimage of boundary of spacefilling curve
Given a continuous map $\gamma$ from $[0,1]$ onto a bounded
contractible subset $S$ of $\mathbb R^2$ such that $S$ contains an open subset of $\mathbb R^2$ which is dense in $S$,
the preimage $\gamma^{...
- 17.9k
1
vote
0
answers
155
views
Study of the class of functions satisfying null-IVP
$\mathcal{N}_u$ : Class of all uncountable Lebesgue-null set i.e all uncountable sets having Lebesgue outer measure $0$.
Let $f:\Bbb{R}\to \Bbb{R}$ be a function with the following property :
$\...
- 307
1
vote
0
answers
125
views
Translation of an article by Šapirovskiĭ
The AMS has a book, Fourteen Papers Translated from the Russian, containing an article by Šapirovskiĭ, "Cardinal invariants in compact Hausdorff spaces", that I would very much like to ...
- 674
1
vote
0
answers
54
views
Closed linear span of compact open subsets of a spectral space
Let $X$ be a spectral space and $KO(X)$ be the set of all compact open subsets of $X$. Identify $KO(X)$ with $\{1_D:D\in KO(X)\}$, where $1_D(u) = 1$ if $u\in D$ and $1_D(u) = 0$ if $u\notin D$.
...
- 2,605
1
vote
0
answers
84
views
Terminology for the property: "Each uncountable disjoint open family is locally countable"
Suppose that a topological space $X$ satisfies the following property
(P): "Each uncountable disjoint open family is locally countable",
where a family $\mathcal U$ of subsets of $X$ is ...
- 505
1
vote
0
answers
97
views
Are Hölder functions between Banach spaces residual in the compact-open topology?
Let $X$ and $Y$ be Banach spaces and let $C(X,Y)$ be the set of continuous functions from $X$ to $Y$ equipped with the topology of uniform convergence on compact sets (i.e. the compact-open topology). ...
- 5,405
1
vote
0
answers
200
views
Question regarding affine fibre bundles
Let $f:X\to Y$ be a morphism of affine varieties such that it is a fibre bundle with fibre $F$. Let $\pi_1(Y)=\Gamma$ be a free group (non abelian) of finite rank and $\pi_1(F)$ is a finite group $G$ ...
- 585
1
vote
0
answers
112
views
What is the topological characteristic of a separable metric space $X$ such that $|kX\setminus X|=\frak{c}$ for any completion $kX$ of $X$?
What is the topological characteristic of a separable metric space $X$ such that $|kX\setminus X|=\frak{c}$ for any completion $kX$ of $X$?
- 1,101
1
vote
0
answers
93
views
What is t-equivalence in function spaces?
In $C_p$-Theory monographs, it is said that two topological spaces $X$ and $Y$ are said to be $t$-equivalent means that $C_p(X)$ is homeomorphic to $C_p(Y)$. Then they also define $u$-equivalences (...
- 31
1
vote
0
answers
46
views
A question related to injective envelope for a system of DGA's
I was trying to read Fine and Triantafillou's paper "On the equivariant formality of Kahler manifolds with finite group action".
They have defined the enlargement at $H$ of a system of DGA's ...
- 1,177
1
vote
0
answers
67
views
Multisymplectic connections and topological invariants
I was wondering if there exists any notion of multisymplectic connection that naturally generalizes symplectic (Fedosov) connections in symplectic geometry.
From symplectic connections, it is well ...
- 405
1
vote
0
answers
110
views
Zeroth homology of the complement of a closed set
Suppose $F$ is a closed set in $\mathbb{R}^n$ with $n>1$.
Are there some known conditions that must be imposed on $F$ so that its complement in $\mathbb{R}^n$ has a finite number of components? ...
- 411
1
vote
0
answers
143
views
End space of non-compact 2-manifolds described with proper rays
I am wondering if the classification of noncompact surfaces given by Ian Richards can be stated with proper rays instead of nested sequences of connected open subsets with compact boundary. I asked ...
1
vote
0
answers
68
views
Proper Morse function on open set
Let $M$ be a compact submanifold with boundary of $\mathbb{R}^n$ of dimension $n$. Let $f:M \to \mathbb{R}$ be a Morse function. Then $f$ is proper. Let $N:=M-bd(M)$. How can I get a proper Morse ...
- 1,177
1
vote
0
answers
43
views
Interleaving in Viennot's Heaps models?
I am looking for past results on interleaving of heaps (in the sense of Viennot). For a very simplified example, suppose I have two pieces, (b1 a1 b1), and (b2 c2 b2), where the letter represents a ...
- 11
1
vote
0
answers
34
views
selection theory for normal non-paracompact domains?
Are there theorems in selection theory without either paracompactness or convexity assumptions ?
That is, a theorem that claims existence of selections for any (perfectly or hereditary) normal spaces, ...
- 317
1
vote
0
answers
150
views
Lifting theorem for finite spaces: replacing perfect normality by normality
In the Lifting theorem for finite spaces (Thm. 3.5, Eric Wofsey, quoted below),
can one relax the condition "$A$ is a closed subset of a perfectly normal $X$" to
"$A\to X$ has the right ...
- 317
1
vote
0
answers
70
views
Distance to set defined as subzero level set of a continuous function
I am searching for strategies on how to prove/disprove that scalar functions "capture" the distance to the subzero level set of the same function. (Or what topics to study to become better ...
1
vote
0
answers
191
views
Questions related to Morse theory
I asked this question long back but could not get a satisfying answer. Starting with a finite CW complex $X$ of dimension $n\geq 3$. I have embedded it inside $R^{2n}$. Now take a tubular neighborhood ...
- 1,177
1
vote
0
answers
79
views
Homotopy type of a general fiber for a polynomial
Define a polynomial $f: \mathbb{C}^2 \to \mathbb{C}$ by $f(x,y)= x(x(2y+1)+1)(x(2y+1)-1).$ The inverse image of zero (i.e. $f^{-1}(0)$) is $\mathbb{C}\cup \mathbb{C}^*\cup \mathbb{C}^*$ (the unions ...
- 1,177
1
vote
0
answers
218
views
How to check a fiber bundle is trivial
Given a smooth fiber bundle $X \to S^1,$ such that the fiber, $F$, is homotopic to $S^2 \vee S^2.$ Is it true that this is always a trivial fiber bundle?
In general, how to check a fiber bundle is ...
- 1,177
1
vote
0
answers
119
views
Inclusion inducing isomorphism at all level except one
Let $V$ is a projective hypersurface of dimension $3$ and $D$ be divisor at infinity of $V$ (assume $D$ has isolated singularities). It is known that the third homology of both $V$ and $D$ are hard to ...
- 1,177
1
vote
0
answers
56
views
Effect on finite transformation semigroup under a particular modification of the generators
The following question arises in connection with problems in automata theory related to the road problem. Let $f_1, f_2: [N] \to [N]$ be maps such that the transformation semigroup $S = \langle f_1, ...
- 695
1
vote
0
answers
70
views
If $X^{\star} \times Y^{\star}$ is $KC$ space, then $X\times Y$ is a $k$-space
The following theorem is found in the article "ON KC AND k-SPACES, A. García Maynez. 15 No. 1 (1975) 33-50"
Theorem 3.5: Let $X, Y$ be topological spaces. If $X^{\star} \times Y^{\star}$ is ...
- 111
1
vote
0
answers
66
views
First isomorphism theorem for inverse semigroups together with v-prehomomorphisms?
In this old paper D. B. McAlister has introduced another class of morphisms for inverse semigroups, called v-prehomomorphisms. For such a morphism $\theta : S \to T,$ instead of preserving the ...
- 1,093
1
vote
0
answers
541
views
Is the set of compact operators closed with the strong topology?
It is well-known that the space of compact operators over Banach spaces is closed within the norm topology.
My question:
Let $X$ be a Banach space.
Considering the strong topology (defined by ...
- 299
1
vote
0
answers
69
views
Topological intuition for the cancellation property of separated maps w.r.t a class of properties of continuous maps
Recall a continuous map is separated if its diagonal is closed. This is equivalent to the fibers being relatively Hausdorff in the total space.
Proposition. Suppose $\mathrm P$ is a class of ...
- 10.5k
1
vote
0
answers
170
views
What is this algebraic object (special case of a semigroup)?
Let $(M,*)$ be a finite semigroup. Further we demand the following:
Zero element: $\exists0\in M \forall m\in M:0*m=0=m*0$.
Left cancelation: $\forall m,n,n'\in M:0\neq m*n =m*n' \Rightarrow n=n'$.
...
- 696
1
vote
0
answers
70
views
Injectivity of post-composition operator
Let $X$, $Y_1,Y_2$, and $Z$ be separable metric spaces. Let $C(X,Y)$ be the topological space of continuous functions from $X$ to $Y$ equipped with its compact-open topologies. Fix a continuous ...
1
vote
0
answers
139
views
Terminology for an kind-of principal fibration
My interest is in topological monoids, but I think the question may make sense (in some fashion) for monoids of sets.
Let $M$ be a topological monoid, and let $X$ be a pointed space that $M$ acts on, ...
- 12.5k
1
vote
0
answers
95
views
Characterization of a topological space by continuous functions
I was reading the paper of Kando about charachterization of topological spaces by some continuous functions. The following generalization came to my mind.
Let $(X,T)$ be a topological space, and $C(X)$...
- 883
1
vote
0
answers
128
views
Is every topological space homeomorphic to a subspace of a locally contractible space?
Is every topological space homeomorphic to a subspace of a locally contractible space?
- 11
1
vote
0
answers
864
views
The "interior" of a convex set?
Consider any compact metric space $S$ which is a convex subset of a vector space. I am trying to see if the following definition is equivalent/close to some well-known definitions.
Consider a convex ...
- 157
1
vote
0
answers
121
views
Why $\beta S$ is not a semigroup when $S$ is a (directed) partial semigroup?
Given a semigroup $(S, *)$ we extend the semigroup operation $*$ of $S$ to a operation $*$ on $\beta S$ (the set of ultrafilters on $S$) defined as
$$
\mathcal{U} * \mathcal{V} = \left\{ A \...
- 59
1
vote
0
answers
77
views
Given a homeomorphism on $\mathbb{R}^3$, can its effects on a compact subset be realized by a homeomorphism that's non-identity only on a compact set?
Let $f_1 \colon \mathbb{R}^3 \to \mathbb{R}^3$ be a homeomorphism, and let $K_1 \subseteq \mathbb{R}^3$ be compact. Does there always exist a homeomorphism $f_2 \colon \mathbb{R}^3 \to \mathbb{R}^3$ ...
- 11
1
vote
0
answers
137
views
Relative compactness... but what is the toplogy?
The following Theorem was described in a text I was reading as a compactness result. The proof is probably too advanced for me but I was just wondering with respect to what topology we have ...
- 111
1
vote
0
answers
139
views
Free monoids on posets
I've suddenly found myself working with some free monoids $F(S)$ in which the set $S$ is a poset, and the order extends to an order $F(S)$, satisfying
if (but not only if) $s_1, s_2, \ldots, s_r, t_1, ...
- 12.5k
1
vote
0
answers
259
views
Spaces homotopy equivalent over the topologist's sine curve
Consider $$T=\left\{ \left( x, \sin \tfrac{1}{x} \right ) : x \in [-1, 0)\cup(0,1] \right\} \cup \{(0,0)\}\subset \mathbb{R}^2$$
with the subspace topology.
Denote $p=(-1, \sin -1), q=(1, \sin 1)\...
- 51
1
vote
0
answers
99
views
topological dimension of inverse limit of compact spaces
Let $X$ be a compact metric space with topological dimension ${\rm dim}(X)>0$. Let $f: X\times X\to X$ be a continuous and surjective map. Define a family of maps $f_n: X^{n+1}\to X^{n}$ for $n\ge ...
- 675
1
vote
0
answers
298
views
Boundary map in Mayer-Vietoris sequence of cohomology
Suppose $M$ is a 3-manifold with connected boundary. Let $T$ be a tangle in $M$, i.e., $T$ is a embedded connected 1-submanifold whose boundary is on $\partial M$ ($T$ is not closed). Moreover, ...
- 673
1
vote
0
answers
119
views
May sequential continuity of a map on compact sets fail to admit extrema?
Let $X$ be a compact topological space. Is there an example of a sequentially upper-semicontinuous function $f: X \rightarrow \mathbb{R}$ that does not admit a maximum point in $X$?
My very rough ...
- 215
1
vote
0
answers
88
views
Are there results on cardinal function using o-tightness?
Recall that a space $X$ has countable $o$-tightness, if for every family $\mathcal U$ of open sets of $X$
and for each $x \in X$ with $x \in \overline{\bigcup \mathcal U}$, there exists a countable ...
- 621
1
vote
0
answers
105
views
Existence of global section, étale map and totally disconnected space
I am trying to show the following result :
Let $Y$ be a totally disconnected space and compact space, $X$ a locally compact space and $p:Y\to X$ a surjective local homeomorphism. Then, there exist ...
- 53