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Questions tagged [general-topology]

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; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

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Convergence in the Hausdorff Metric

I want to prove that if $A_n$ is a sequence of nonempty compact subsets of the metric space $S$ and $A$ is a subset of $S$ such thant $A_n$ converges to $A$ in the Hausdorff metric, then $A_n = \{ x: \...
Rubén Sales Castellar's user avatar
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Triangulations of manifolds are non-branching

Let $X$ be an $n$-manifold and let $K$ be a triangulation of that manifold. I am looking for a proof of the fact that $K$ is non-branching, which means: There is no simplex $S \in K$ of dimension $n-1$...
shuhalo's user avatar
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2 votes
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26 views

Short exact sequences of orthonormal frame bundles

Given an orthonormal $l$-frame bundle $V_l(TM)\rightarrow M$ for a smooth (oriented) $d$-manifold $M$, there are short exact sequences on homotopy groups $\mathbb{Z}\rightarrow \pi_{d-l}(V_l(TM))\...
Tiana's user avatar
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0 answers
24 views

Function property on balls and scaling

Assume $X$ is a compact metric space and $Y$ is a metric space. Assume $F:X\rightarrow Y$ is a map. Fix $L\geq 1$. Suppose for each $x\in X$, there exists an $\epsilon_x>0$ such that for every $w_1,...
monoidaltransform's user avatar
-1 votes
1 answer
51 views

How to Prove a Set is Closed in $\mathbb{R}^2$ using the product topology definition?

Question: I'd like to prove that the set $A= \left\{ (x,\frac{1}{x}): x \in \mathbb{R} \setminus \{0\} \right\}$ is closed in $\mathbb{R}^2$ endowed with the product topology. Although it can be shown ...
bayes2021's user avatar
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1 vote
1 answer
31 views

Continuous injection to manifold with boundary

Let $M^n$ be an n dimensional manifold with boundary. Assume $U$ is open in $M^n$. Let $f:U\rightarrow M^n$ be a continuous injection open map. Then show that $f(U\cap \partial{M})=\partial{M}$ I am ...
monoidaltransform's user avatar
1 vote
0 answers
29 views

A lemma of Cielsielski

Reading section 2 of this paper '' A century of Sierpinski–Zygmund functions'' (Ciesielski, K.C., Seoane-Sepúlveda, J.B.. Rev. R. Acad.Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 113(4), 3863–3901 (...
sabrina's user avatar
  • 11
3 votes
1 answer
45 views

Well-definedness of the projection associated to the sheaf of germs of a presheaf

I'm currently reading Izu Vaisman's Cohomology and differential forms ($1973$) having never studied sheaf theory before, so I will briefly write down the definitions in case they don't match with ...
Bruno B's user avatar
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2 votes
1 answer
25 views

Must scattered spaces with points $G_\delta$ be locally countable?

By M. E. Gewand, “The Lindelöf degree of scattered spaces and their products,” Journal of the Australian Mathematical Society (Series A) 37 (1984), 98–105, we have: Theorem. Every Lindelöf scattered ...
Steven Clontz's user avatar
6 votes
1 answer
375 views

Is a continuous function a generalization of an adjunction?

In his recent blog post, Bartosz Milewski notes that we can define a continuous function on two topological spaces $(X, \mathcal{T}_X), (Y, \mathcal{T}_Y)$ as a pair of functions $(f,g)$ with $f:X\to ...
LandOnWords's user avatar
0 votes
0 answers
40 views

Given a metric $d$, is continuity of $f$ absolutely essential for the composite function $f \circ d$ to be a metric (equivalent to $d$)? [closed]

Let $d$ be a metric on a nonempty set $X$, and let $f \colon [0, +\infty) \longrightarrow [0, +\infty)$ be a function such that (i) $f(0) = 0$, (ii) $f(r+s)\leq f(r) + f(s)$ for all $r, s \in [0, +\...
Saaqib Mahmood's user avatar
1 vote
1 answer
31 views

Specific construction of loop in Union of path connected spaces

I have been trying to get my head around this question and its solution. The (easier and more straightforward) solution using the Lebesgue Number Lemma is greatly explained here: Prove that a space is ...
manifold97's user avatar
1 vote
0 answers
27 views

(Number of) 2-sheeted coverings of $\mathbb{R}P^2 \times S^1$

I recently came across a question, asking for the number of 2-sheeted covering maps of $\mathbb{R}P^2 \times S^1$ (up to equivalence), and I really have no clue. There are at least three for sure, but ...
Batixx's user avatar
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69 views

What are the "standard" topology for a space of random variables?

Suppose $\mathbb X$ is the space of random variable over $(S,\Sigma, \mu)$ with finite support. What are some commonly used topology for $\mathbb X$? Reason of asking: I am reading a paper which uses ...
dodo's user avatar
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0 answers
47 views

Is every subset of C^n whose coordinates are real not open? [closed]

Consider the space $\mathbb{C}^n$, where each point is represented by $z = (z_1, z_2, \ldots, z_n)$ with $z_j \in \mathbb{C}$ for $j = 1, 2, \ldots, n$. A point $z \in \mathbb{C}^n$ has real ...
jujumumu's user avatar
  • 133
0 votes
0 answers
42 views

Continuity of radii of balls function

Assume $X$ is a nice enough metric space. Assume $U$ is an open set in $X$. Is it true that there exists a continuous function $f:U\rightarrow (0,\infty)$ such that $B_{u}(f(u))\subseteq U$ for all $...
monoidaltransform's user avatar
3 votes
0 answers
35 views

Homology and Brouwer degree theory over: the quotient space obtained of the suspension of $S^1 \vee S^1$ identifying its endpoints.

Hi, everyone! I think my solution isn't the more economic. Actually, I'm not satisfied, and to be honest I think I miscalculated the celular homology of $$ X= \dfrac{(S^1 \vee S^1)\times[-1,1]}{(S^1\...
A-Train's user avatar
  • 31
4 votes
1 answer
129 views

Is a topological abelian group with no open subgroup connected?

You may assume that the groups we consider are all Hausdorff for the purpose of this question: Clearly, the implication converse to the one in the question holds, as all open subgroups are also ...
Luca Marchiori 's user avatar
0 votes
1 answer
55 views

Proof of $f(\overline{S}) = \overline{f(S)}$

I know that $f(\overline{S}) = \overline{f(S)}$ if and only if $f: X \rightarrow Y$ is an homeomorphism, you can find a proof here. I am trying to establish a proof of the $\impliedby$ part, but I don'...
ToT's user avatar
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1 vote
0 answers
39 views

Find compactification of $\mathbb{R}$ which has a subset homeomorphic to $\mathbb{R}^2$

Consider $X=\mathbb{R}$ with the standard topology. How can I find a compactification $Y$, such that $Y$ is (of course) compact, hausdorff and has a subset which is homeomorphic to $\mathbb{R}^2$? ...
FreeZe's user avatar
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0 votes
0 answers
35 views

Can any open set in $\mathbb{R}^d$ be countably union of closed sets

I've already know that $\{B(x,r):x\in\mathbb{Q}^d,r\in \mathbb{Q}\} $ is an countable base of $\mathbb{R}^d$. Intuitively, I wonder that can an open set $\Omega\subset \mathbb{R}^d$ be countably union ...
Hải Nguyễn Hoàng's user avatar
-2 votes
0 answers
23 views

For $\overline{\{\xi \}} = X'\subseteq X$ with $\xi \in S \subseteq X$, $\{\xi\}$ is dense in $S \cap X'$?

It seems simple question and I wonder whether there is a mistake for proof. Let $S\subseteq X$ be a subspace of topological space $X$. $ \overline{\{ \xi \}}=X' \subseteq X$ be a subspace with generic ...
Plantation's user avatar
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0 votes
0 answers
22 views

$H:M\rightarrow \mathbb{R}$ is a continuous map from top space $M$. Show that $H^{-1}(e)$ divides $M$ where $e\in \text{int}(H(M))$

Let $M$ be a topological space and $H:M\rightarrow \mathbb{R}$ be continuous and surjective. Suppose $e\in \text{int}(H(M))$. Then show $H^{-1}(e)$ divides $M$; that is, $M\setminus H^{-1}(e)$ has ...
Ali's user avatar
  • 356
1 vote
0 answers
71 views

Issues with a quotient topological space [closed]

Let $X=S^1$ the unit circle and take $p\in X$ and define $A=X\setminus \{p\}$. Consider the quotient space $Y=X/A$, that means $x\sim y$ iff $x,y\in A$. Is $Y$ homotopic equivalent to $X$? My problem ...
Sigma Algebra's user avatar
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0 answers
36 views

polyhedral space questions

In class, as a recommendation, we were recommended to read https://www.maths.gla.ac.uk/~mpowell/Stallings%20Polyhedral%20topology%20typed.pdf. On page 5, the author defines euclidean polyhedrons as (...
monoidaltransform's user avatar
-1 votes
0 answers
84 views

Show that $A=\{ (x, \frac{1}{x}): x \in \mathbb{R} \setminus \{0\} \}$ is closed in $\mathbb{R}^{2}$ [duplicate]

I'd like to show that the set $A= \left\{ (x, \frac{1}{x}): x \in \mathbb{R} \setminus \{0\} \right\}$ is closed in $\mathbb{R}^{2}$ endowed with the product topology. It can be shown that $A$ is ...
bayes2021's user avatar
  • 633
1 vote
0 answers
28 views

Locally compact Hausdorff covering space

I am trying to prove the following conjecture: Let $\pi : E\to B$ be a covering map. If $E$ is locally compact Hausdorff, then so is $B$. (The converse is known to be true, cf. Exercise 6 in Section ...
Nick F's user avatar
  • 1,249
-1 votes
1 answer
60 views

Understanding proof of existence of Schreier transversals

I'm trying to understand the proof in the image attached. I'm fine with everything in the first paragraph of the proof. But I don't get why the stuff in the last sentence holds. Why is H freely ...
Keven McFlurry's user avatar
4 votes
1 answer
378 views

Is this "continuous" function really continuous?

Let's assume $n \ge 2.$ Suppose $f:\Bbb R^n \times \Bbb R \to \Bbb R^{n+1}$ has the form $$ f(x,t) = (\phi_t(x),t), $$ where $\phi_{t_0}:\Bbb R^n \to \Bbb R^n$ is a continuous function for each fixed $...
BigbearZzz's user avatar
  • 15.3k
-2 votes
0 answers
31 views

Prove that a product of first-countable topological spaces is first-countable iff only countably many of them don't have the trivial topology [closed]

I need to show that if $\{X_\alpha\}_{\alpha \in \Lambda}$ are all first-countable spaces, then $\prod_{\alpha \in \Lambda} X_\alpha$ is first-countable if and only if there exists a $\Lambda' \...
NivGeva's user avatar
  • 29
1 vote
0 answers
101 views

Do you know if this theorem has a name?

I'd like to ask if the following theorem has a name. Let $M$ be a smooth $n$-dimensional manifold, $\mathcal{P}$ a family of open subsets of $M$ satisfying the following: $(1)$ $\emptyset \in \mathcal{...
Paz's user avatar
  • 21
0 votes
0 answers
47 views

Find a topological manifold that is not Hausdorff and not locally compact

Define a topological manifold as a space locally homeomorphic to $\mathbb{R}^n$. Find a topological manifold that is not Hausdorff and not locally compact. (Hint:Consider $\mathbb{R}$ with "extra ...
Ali's user avatar
  • 356
0 votes
0 answers
33 views

Smoothness of a function is well-defined for functions between smooth manifolds

Definition: A function $F: M\rightarrow N$ smooth at $p\in M$ if there is a chart $(U,\phi)$ about $p$, a chart $(V,\psi)$ about $F(p)$ such that $F(U) \subset V$ and the map $\psi \circ F \circ \phi^{...
Jeffrey Jao's user avatar
0 votes
0 answers
48 views

Suppose $M/G$ is a smooth quotient manifold and $N$ is a $G$-invariant submanifold of $M$. Then is $N/G$ a submanifold of $M/G$?

Let $M$ be a smooth manifold equipped with a, not necessarily proper, smooth Lie group action $G$. Suppose $M/G$ is a smooth quotient manifold. That is, there exists a smooth structure on the quotient ...
Spencer Kraisler's user avatar
1 vote
1 answer
63 views

Intermediate Value Theorem on a Tree

Suppose we have a finite tree $T$ and a continuous function $f:[0,1]\to |T|$. By $|T|$ I mean the underlying space which we can think of as a subspace of $\mathbb{R}^2$. It seems intuitively obvious ...
H_R's user avatar
  • 443
1 vote
2 answers
39 views

In a regular space with a $\sigma$-locally finite network, is every closed set a $G_\delta$?

In a regular space with a $\sigma$-locally finite network, is every closed set a $G_\delta$? I'm inclined to think so based upon the basic argument of Theorem 2.8 here, but there are a few moving ...
Steven Clontz's user avatar
0 votes
0 answers
33 views

Question regarding the fundamental group of a quotient space.

$\mathbf{The \ Problem \ is}:$ Let $X:= S^2\times [-1,1]$ generated by $(x,t)\sim (-x,t)$ for $x\in S^2$ and $t\in \{-1,0,1\}.$ Show that $\pi_1(X/\sim,x_0)$(at any base point $x_0$) is isomorphic to $...
Rabi Kumar Chakraborty's user avatar
0 votes
1 answer
26 views

Verifying proof that in a first countable space if $u_n\to u$ and $f(u_n)\to f(u)$, then $f$ is continuous

Let $S$ and $T$ be topological spaces with $S$ first countable and $\varphi: S \to T$. Then $\varphi$ is continuous iff for every sequence $\{u_n\}$ converging to $u$, $\{\varphi(u_n)\}$ converges to $...
Raffaella's user avatar
  • 115
4 votes
1 answer
77 views

Equivalence of two strengthenings of open neighborhood

I'm currently pursuing a project (about Markov conditions in Minkowski spacetime) in which a certain equivalence between two strengthenings of open neighborhood plays an important role. Say that set $...
Jens's user avatar
  • 119
2 votes
0 answers
52 views

does every open subset of a locally compact space contain a compact set? [closed]

Let $X$ be a locally compact Hausdorff space with basis $\mathcal{O}$. Let $U$ be a non-empty basis element. Does $U$ necessarily contain a non-empty compact neigbhourhood $K$ containing some $x\in U$?...
Panini's user avatar
  • 357
0 votes
0 answers
43 views

Packing tiny open sets into a large open set

The questions Let $n$ be a positive integer, let $\lambda$ be the Lebesgue measure of $\mathbb R^n$, let $B$ and $U$ be two open subsets of $\mathbb R^n$ such that $B$ is bounded, $\lambda(B)=1$, and $...
Pierre-Yves Gaillard's user avatar
0 votes
0 answers
66 views

Is there another topology on a given set which makes closures equal?

In a research article about Lindelöfness, I saw a topological space. Let $A=\{a,b\}$, $B=\{c_{i}: i<\omega_{1}\}$ and $C=\{a_{ij}: i<\omega_{1}, j\in\mathbb{N}\}\cup\{b_{ij}: i<\omega_{1}, j\...
Woodx's user avatar
  • 185
2 votes
1 answer
38 views

If $M$ is a connected topological manifold and $p,q \in M^{o}$, then there exists a homeomorphism which maps $p$ to $q$.

I am stuck in constructing a explicit homeomorphism $f$(say) from a connected manifold $M^{o} = M \backslash \delta M$ to itself such that for $p,q \in M^{o}$ then $f(p) = q$. I know that that since $...
Dwaipayan Sharma's user avatar
2 votes
0 answers
76 views

Prove that the family of neighborhoods of $\Bbb N$ in $\Bbb R$ does not have a countable basis. [duplicate]

Let $X$ be a topological space and $A$ a nonempty subset of $X$. A subset $V$ of $X$ is called a neighborhood of $A$ if there exists an open subset $U$ of $X$ such that $A\subset U \subset V$. The set ...
noname1014's user avatar
  • 2,533
0 votes
1 answer
39 views

Can a function over a domain with holes have a derivative of zero everywhere but fail to be constant? [duplicate]

I've been reviewing some real analysis to prepare for topology and I got myself into a pickle while trying to understand disconnected sets and holes. Here is some reasoning that arrives at a weird ...
Cole Hillyer's user avatar
0 votes
0 answers
42 views

Can we pass between sheaves of suitable type and fibre bundles (not ! etale bundles)?

Today I was forced to think about vector bundles (on manifolds), but in the context of a textbook "otherwise" focused on sheaf theory. I'm interested to what extent this really is an "...
FShrike's user avatar
  • 42.6k
0 votes
1 answer
38 views

Compactess of a set in $\mathbb{R}^d$ defined as the union of compact sets

Let $f:[a,b]\to \mathbb{R}^d$ be a function of class $C^1$. Let us consider the following set: $$ \mathcal{A}:=\bigcup_{x\in [a,b]}\{y\in \mathbb{R}^d, \quad ||y- f(x)||\leq 1/2 \} $$ I think that ...
hanava331's user avatar
  • 109
2 votes
1 answer
30 views

Non-proper smooth embedding

Does all smooth embedding $\iota: S\to M$ is proper if $\dim S < \dim M$? I thought then the slice criterion implies $S$ is locally closed, so closed in $M$. But this is equivalent to saying that ...
okabe rintarou's user avatar
-1 votes
1 answer
15 views

Composition of asymmetric contraction mappings [closed]

Let $(M,d)$ and $(N,q)$ be metric spaces. The operator $T:M\longrightarrow N$ is contractive in the sense that $q(T(m_1),T(m_2)) \leq c d(m_1, m_2)$ for some $c\in [0,1)$. Similarly, the operator $J:N\...
phil's user avatar
  • 162
2 votes
0 answers
62 views

When is quotient of $S^n$ by $\mathbb{Z}_2$ action $\mathbb{R} \mathbb{P}^n$?

I'm wondering when does a quotient of $S^n$ by a free $\mathbb{Z}_2$ action give $\mathbb{R} \mathbb{P}^n$. I.e. suppose $$ F: S^n \to S^n$$ $$F^2(x) = x$$ and $F$ does not fix any point. Then is $S^n ...
JMK's user avatar
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