Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; ...
18
votes
0answers
147 views
Does $X\times S^1\cong Y\times S^1$ imply that $X\times\mathbb R\cong Y\times\mathbb R$?
This question came up in a recent video series of lectures by Mike Freedman available through Max Planck Institut's website. He proves the "difficult" converse direction, that $X\times \mathbb R\cong ...
16
votes
0answers
137 views
If $S\times\mathbb{R}$ is homeomorphic to $T\times\mathbb{R}$, and $S$ and $T$ are compact, can we conclude that $S$ and $T$ are homeomorphic?
If $S \times \mathbb{R}$ is homeomorphic to $T \times \mathbb{R}$ and $S$ and $T$ are compact, connected manifolds (according to an earlier question if one of them is compact the other one needs to be ...
13
votes
0answers
203 views
Continuity of a function to the integers
I am trying to prove that in $\mathbb{Z}$ with co-finite topology the only path-connected components are the singletons.
(I reckon that) showing that
"if a function $\gamma : [0,1] \to ...
13
votes
0answers
349 views
Shrinking Group Actions
Suppose $H\subset G$ is a subgroup of a topological group $G$, and $Y\subset X$ is a subspace of a topological space $X$. Suppose we are given a continuous group action $\rho : G\times X\rightarrow X$ ...
13
votes
0answers
595 views
Lifting local compactness in covering spaces
Since the total space of a cover is locally homeomorphic to the base space, local topological properties (like local (path) connectedness, T1 etc.) lift from the base space to the total space. The ...
13
votes
0answers
202 views
What are the attaching maps for the real Grassmannian?
The Grassmannian $G_n(\mathbb{R}^k)$ of n-planes in $\mathbb{R}^k$ has a CW-complex structure coming from the Schubert cell decomposition.
The study of characteristic classes tells us that these ...
12
votes
0answers
813 views
When is the image of a null set also null?
It is easy to prove that if $A \subset \mathbb{R}$ is null (has measure zero) and $f: \mathbb{R} \rightarrow \mathbb{R}$ is Lipschitz then $f(A)$ is null. You can generalize this to $\mathbb{R}^n$ ...
11
votes
0answers
208 views
Understanding Alexandroff compactification
Is the Alexandroff one-point compactification of a locally compact Hausdorff space ($\mathbf{LCHaus}$) a functor to the category of compact Hausdorff spaces ($\mathbf{CHaus}$)? It seems to me that one ...
10
votes
0answers
256 views
Differential forms on fuzzy manifolds
This post will take a bit to set up properly, but it is an easy read (and most likely easy to answer); in any event, please bear with me.
Question
In the usual setting of open subsets of ...
9
votes
0answers
91 views
Restrictions of null/meager ideal
Let I denote the null/meager ideal on reals. Is it consistent that for any pair of non null/meager sets A and B, there is a null/meager preserving bijection between A and B? In particular, is this ...
9
votes
0answers
107 views
How much do idempotent ultrafilters generate in terms of semigroups?
It is known that the set of ultrafilters on, say, the natural numbers $\mathbb{N}$, can naturally be endowed with the structure of a compact topological left semigroup (which fails to be anything ...
9
votes
0answers
158 views
Is there a similar concept for a sigma algebra like a base for a topology?
For both a sigma algebra and a topology, we can talk about their generators.
For a topology, a base is a special generator only using union, which is a useful concept in topology.
In parallel ...
8
votes
0answers
48 views
Are there more embeddings $U(2) \hookrightarrow SO(4)$?
It is easy to prove that $SO(4)$ acts transitively and freely on $S^2$ with fiber $U(2)$. Therefore, we can identify each point of $S^2$ with a particular embedding $U(2) \hookrightarrow SO(4)$.
My ...
8
votes
0answers
326 views
Compactification of Manifolds
It is known that for any locally compact Hausdorff space X, we can define a Hausdorff one-point compactification containing X.
In the case of the (differentiable) manifold $\mathbb R^n$ this one-point ...
7
votes
0answers
80 views
Existence of a map in a Hilbert space
Let $H$ be an infinite-dimensional Hilbert space, $B$ be its unit ball: $B=\{x\in H: \, \|x\|\leq 1\}$.
Does there exist a continuous map $f:H\to H$ such that $f(f(x))=x$ $\forall x\in H$, $f$ has no ...
7
votes
0answers
112 views
Path Connectedness and continuous bijections
Mathoverflow.
Are there any two topological spaces $X$ and $Y$ such that they are path connected and such that there exist continuous bijections $X\rightarrow Y$ and $Y\rightarrow X$, but and yet ...
7
votes
0answers
128 views
What properties are preserved under a measurable mapping?
Although in an abstract category the morphisms are not explicitly defined, in a concrete example (model theory?), morphisms are (always/usually?) mappings that preserve some properties.
In the ...
7
votes
0answers
98 views
Open map which “almost fixes” the boundary of an open ball
We have a continuous function $f:\bar{B}\to\mathbb{R}^n$, where $\bar{B}=\{x\in\mathbb{R}^n:\|x\|\le 1\}$, such that if $\|x\|=1$ then $\|f(x)-x\|<\epsilon$, for a fixed $\epsilon\in(0,1)$. We have ...
7
votes
0answers
141 views
Is there an abelian cat of topological groups?
There are lots of reasons why the category of topological abelian groups (i.e. internal abelian groups in $\bf Top$) is not an abelian category. So i'm wondering:
Is there a "suitably well ...
7
votes
0answers
319 views
Mistake in Dugundji chapter IX section 11 example 3?
A family of pseudometrics defined on a set gives rise to a uniform structure on that set. Moreover (up to uniform equivalence, anyway) every uniform structure arises this way. Let $A$ and $B$ be ...
6
votes
0answers
80 views
Visualize Fourth Homotopy Group of $S^2$
I know $\pi_4(S^2)$ is $\mathbb{Z}_2$. However, I don't know how to visualize it. For example, it is well known that $\pi_3(S^2)=\mathbb{Z}$ can be understood by Hopf Fibration. Elements in ...
6
votes
0answers
67 views
“Cutting out” loops from paths
Given a topological space $X$ and a path $p : [0, 1] \to X$ such that $p(0) \neq p(1)$, does there always exist an injective path $p' : [0,1] \to \operatorname{im}p$ such that $p'(0) = p(1)$ and ...
6
votes
0answers
71 views
topological group operation vs homotopy group operation
Let $X$ be a topological group. Let $\tau_1$ and $\tau_2$ representing elements of $\pi_n(X)$. Is it true that
$$ [\tau_1] [\tau_2] = [\tau_1 \tau_2] $$
in $\pi_n(X)$?,
where of course "$[\tau_1] ...
6
votes
0answers
105 views
What are 'weak' forms of Urysohn's lemma, which do not require choice?
Reference: http://web.mat.bham.ac.uk/C.Good/research/pdfs/horror.pdf
It is well known that the original proof of Urysohn's lemma uses a choice principle. (DC)
What are weak forms of the Urysohn's ...
6
votes
0answers
202 views
Topological necessary and sufficient condition for tightness
Recall the definition of tightness for a probability measure $\mathbb P$ on the Borel $\sigma$-algebra of a metric space $(S,d)$:
For each $\varepsilon>0$, we can find a compact subset $K$ of ...
6
votes
0answers
85 views
Piecewise Affine Bijections of $\mathbb{R}^n$
I have a min-max function $f:\mathbb{R}^n\to\mathbb{R}^n$ of the form $$f(x) = \min_{i=1,\dots,n}\max_{j=1,\dots,n}(\alpha_{ij}^Tx + \beta_{ij})\quad\text{where each } \alpha_{ij}\in ...
5
votes
0answers
57 views
Characterization of open maps
I'm looking for different but equivalent definitions of the concept of open map. So Let $X,Y$ be topological spaces and $f:X\longrightarrow Y$ a function, not assumed to be continuous. I conjectured ...
5
votes
0answers
89 views
Is dependent choice necessary to prove every perfect compact Hausdorff space is uncountable?
The answer to Cardinality of a locally compact space without isolated point shows that AC is required to show that if $X$ is a compact Hausdorff space with no isolated points then $|X| \ge ...
5
votes
0answers
34 views
$M$ is compact, non-empty, perfect, and $M \cong M \times M$. Must $M$ be homeomorphic to the Cantor set, the Hilbert cube, or some combination?
Assume that $M$ is compact, non-empty, perfect, and homeomorphic to its Cartesian square, $M \cong M \times M$. Must $M$ be homeomorphic to the Cantor set, the Hilbert cube, or some combination of ...
5
votes
0answers
87 views
I think a definition is wrong in “Model Categories” by Hovey.
I am working through the book "Model Categories", by Mark Hovey, and have a doubt about a definition given there. At the beginning of page 50 we read:
Define a map $f:X\rightarrow Y$ to be a ...
5
votes
0answers
32 views
A pseudocompact space with $G_\delta$-point
Is evert $T_2$ pseudocompact space with $G_\delta$-points always first countable? Does there exist a counterexample?
Thanks ahead.
5
votes
0answers
91 views
Proving that a set is closed in $L^2(\mathbb{R})$
this is my first question here so i hope i don't do anything wrong. Excuse any spelling or grammar mistakes, english isn't my mother tongue.
I'm reading this paper for my bachelor thesis and have ...
5
votes
0answers
115 views
Proof that the set of doubly-stochastic matrices forms a convex polytope?
Does the set of all doubly-stochastic matrices form a convex polytope? In general, I wonder how the proofs of convexity and geometry can be established for sets of matrices of this kind? Anything to ...
5
votes
0answers
58 views
Homological definition of orientation at a boundary point?
For a topological manifold $M^m$, an orientation at a point $x \in M$ can be defined as a choice of generator for $H_m (M, M-x)$. For a topological manifold with boundary this definition still makes ...
5
votes
0answers
148 views
Baire sets of $X$ possess the required Cartesian product property
Let $X=X_{1}\times X_{2}$ is locally compact space, and define $$E=\{E_{1}\times E_{2}\;|\; E_{i}\; \text{is a Borel set in}\; X_{i}\; ,\; \text{for}\; i=1,2\}$$ Now why the Baire sets of $X$ are in ...
5
votes
0answers
72 views
Topological Space in which every compact subset is metrizable
Is there an (more or less) established name for the class of topological spaces in which every compact subset is metrizable? This is true for example in (LF)-spaces (inductive limits of ...
5
votes
0answers
161 views
How to prove a manifold is diffeomorphic to Euclidean space?
Problem is this: suppose a manifold
$$M=\bigcup_{n\in\mathbb{N}} U_n,$$
where each $U_n$ is diffeomorphic to Euclidean space, and $U_n$ is contained in $U_{n+1}$. Then please show that $M$ is ...
5
votes
0answers
106 views
compact-open metrizability
Given topological spaces $X$ and $Y$ the set $C(X,Y)$ of all continuous functions $f:X\to Y$ becomes a topological space with the compact-open topology (that is the topology generated by the sets ...
5
votes
0answers
310 views
topology puzzle - without cut the rope, separate two rings
hello I wonder whether this puzzle is possible to solve.
if possible, what kind of thing should I learn to solve this?
the problem is make left one to right one without cut the rope
only stretch and ...
5
votes
0answers
74 views
Unicoherence of non-euclidean spaces
My question concerns the notion of unicoherence, which is a property that a topological space may or may not have. The definition (from Wikipedia) is:
"A topological space $X$ is said to be ...
5
votes
0answers
134 views
Are these sets in $\mathbb{R}$ open and/or closed?
In $\mathbb{R}$, are these sets open? Are they closed?
$A = \{\frac{1}{n} : n \in \mathbb{N}\}$
$B = A \cup \{0\} $
$[0, 1)$
My thoughts:
$A$ is not open as if we have an open ball with $r > ...
5
votes
0answers
86 views
When is $\{ x | f(x) \le 0\}$ path-connected?
I'm trying to determine the conditions on $f : \mathbb{R}^n_{\ge 0} \to \mathbb{R^n}$ under which $\{ x | f(x) \le 0\}$ is path-connected. We can assume that $f$ is continuous and concave (i.e. for ...
5
votes
0answers
235 views
A fiber bundle over Euclidean space is trivial.
What's the easiest way to see this? The only thing I could think to do was try to patch together trivializations. I couldn't find a way to make that work. Thank you!
edit: For the record, here's why ...
5
votes
0answers
111 views
Question about proof of Tychonoff-Alaoglu
I'd like to check that I understand the proof in full detail. Can you tell me if the following is correct? Thanks for your help.
Claim: The closed unit ball $B_{\|\cdot\|_{op}}(0,1)$ in $X^\ast$ is ...
5
votes
0answers
71 views
Continuous choice of basis for subspaces
Consider the flag variety (or flag manifold, depending on who you are) $V=\mathrm {Fl} (3,\mathbb C)$ of complete flags of subspaces of $\mathbb C^3$. That is, an element of M is a tuple (L , P) ...
5
votes
0answers
232 views
Generalizations for Tietze's extension theorem.
Tietze's extension theorem says:
''If $A$ is a closed subset of $X$ a normal space, and $f:A\to \mathbb{R}$ continuous, then we can extend $f$ to a continuous function $g:X\to \mathbb{R}$."
I know ...
5
votes
0answers
182 views
Definition of Reshetikhin-Turaev TQFT
I am studying Reshetikhin-Turaev TQFT. In their paper or in the book " Quantum invariants of knots and 3-manifolds", they first define an invariant $\tau(M)$ for a closed orientable 3-manifold $M$ and ...
5
votes
0answers
202 views
Points in the plane at integer distances
Does there exist a set of $n$ points $p_1,p_2,...,p_n$ in the plane, all at mutual integer distances to each other, and an $e>0$, such that the following statement holds:
For all $a,b$ with ...
5
votes
0answers
139 views
Connected subspaces question
Suppose that C, D are connected subsets of a topological space T such that $\bar{C} \cap \bar{D} \neq \emptyset$. Is it true that $C \cup D$ is necessarily connected?
I think I have a counter example ...
5
votes
0answers
190 views
Are Arzelà–Ascoli theorems results of similar theorems on normed spaces, metric spaces or other spaces?
From Wikipedia, two generalizations of the Arzelà–Ascoli theorem are
Let $X$ be a compact Hausdorff space. Then a subset $F$ of $C(X)$,
the set of real-valued continuous functions on X, is ...
