Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

2
votes
2answers
15 views

Is the sequence $( d(x_{n},y_{n}))$ convergent if $X = \mathbb{R}$ with standard topology. [duplicate]

Let $(X,d)$ be a metric space and $ ( x_{n} )$ , $ ( y_{n} )$ convergent sequences in $X$. Is the sequence $( d(x_{n},y_{n}))$ convergent if $X = \mathbb{R}$ with standard topology. I am having ...
1
vote
0answers
12 views

If $H_1 \subset H_2 \subset G$ and $G/H_2,\ H_2/H_1$ are compact then $G/H_1$ is compact.

I'm trying to solve the following exercise: Exercise: Let $G$ be a topological group and $H_1 \subset H_2\subset G$ closed subgroups of $G$. Show that if $G/H_2$ and $H_2/H_1$ are compact, then $G/...
0
votes
0answers
17 views

Cone and unit circle

I was curious about the following ideas and I would appreciate any help or guidance on this: 1) How would I define a unit circle in a euclidean cone where the set of points are a unit from the cone ...
0
votes
2answers
14 views

A net $(x_\alpha)$ has an accumulation point $x$ iff some subnet converges to $x$

I'm looking at the answer to this old question Accumulation point in topological space problem I figured out the backwards direction on my own, but I'm a little confused about how the subnet is being ...
0
votes
1answer
22 views

If some sequence $(x_{n})$ is convergent in $X$ show that $(x_{n})$ is stationary (eventually constant).

Let $X$ be non-empty set and $ T = \{ V \subseteq X : card ( X \backslash V) \leq \aleph_{0} \} \cup \{\emptyset\}$ topology on $X$. If some sequence $(x_{n})$ is convergent in $X$ show that $(x_{n})$...
3
votes
1answer
30 views

Does a topology with countable elements imply that the topology is second countable?

Does a topology with underlying space having countable elements imply that the topology is second countable? Today, the above question come into my mind. This seems very intuitive to me, but I am not ...
0
votes
0answers
15 views

Separately continuous implies continuous

I don't understand the proof of the following result in Kechris' "Classical Descriptive Set Theory", pp. $62$ Theorem $(9.14)$ Let $G$ be a group with a topology that is metrizable and Baire, s.t....
0
votes
1answer
22 views

Topological spaces - Meager in itself

Hello someone know examples of topological spaces of first category in themselves (meager in itself), that are not countable? Note that the Cantor set is nowhere dense in $\mathbb{R}$ but it is not ...
0
votes
1answer
9 views

Prove any subnet of a convergent net is convergent

Let $f:J \rightarrow X$ be a net in $X$, let $f(\alpha) = x_\alpha$. If $K$ is a directed set and $g:K \rightarrow J$ is a function such that $i \leq j \rightarrow g(i) \leq g(j)$ and $g(K)$ is ...
0
votes
0answers
8 views

Vector fields and cones

I have a euclidean cone, $X$, with some cone point $P$. I assume that a vector field on $X - P$ is parallel if an isometry that carries an open set $X - P$ to $\mathbb R^2$ carries the vector field ...
1
vote
0answers
17 views

Orbit space of $\mathbb{Z}_2$-action on the torus $T^2$

Let $\mathbb{Z}_2$ be the group $\{1,-1\}$. I want to construct an action of $\mathbb{Z}_2$ on the torus $T^2$ such that the orbit space is homeomorphic to the sphere $S^2$. Could anyone give some ...
1
vote
0answers
17 views

A question about space quotient and homeomorphism [duplicate]

Let $X$ and $Y$ be topological space and let be $\sim_X$ and $\sim_Y$ the corresponding equivalence relations on $X$ and $Y$. Let $f\colon X\to Y$ an homeomorphism, suppose we are in the following ...
1
vote
1answer
40 views

Topology: continuity

I am not sure how to approach the question. Let $X=\{a,b\}$ and define two topological spaces on $X$ by $\tau_1=\{\emptyset, \{a\}, X\}$ $\tau_2=\{\emptyset, \{b\}, X\}$ Is a function given by $f:(...
2
votes
2answers
37 views

A sequence $ (x_{n})$ is convergent in $X$ $\iff$ $ (x_{n})$ is a stationary sequence. Does $X$ have to be discrete?

Let $X$ be a topological space so that a sequence $ (x_{n})$ is convergent in $X$ $\iff$ $ (x_{n})$ is a stationary sequence. Does $X$ have to be discrete? Does topology $ T = \left\{ X \subset \...
0
votes
0answers
11 views

Existence of a particular complex atlas (and nested precompact sets)

Let $X$ be a complex topological manifold. Can I always find an atlas $A=\{\psi_i:{U_i\rightarrow V_i\subseteq \mathbb{C},i\in \mathbb{N}}\}$ so that $U_i\subseteq U_{i+1}$ (with $\{U_i\}$ an open ...
0
votes
1answer
37 views

Prove that $\{x^{(j)}\} \to x \in X$ if and only if $\{x_k^{(j)}\} \to x_k \in X_k$ for $k=1,2$.

Let $(X_k,d_k)$ for $k =1,2$ be metric spaces. Let $d$ be a metric on $X = X_1 \times X_2$ such that the open subsets of (X,d) are precisely the unions of sets of the form $U_1 \times U_2$, where $U_k$...
1
vote
3answers
60 views

Find the interior $\{(x,y): 0 < x^2 + y^2 < 1\}$.

Find the interior $A = \{(x,y): 0 < x^2 + y^2 < 1\}$. I assume the metric is the standard Euclidean metric. I know that the int$(A) = A$, but I don't know how to prove it. I can gather that the ...
1
vote
1answer
29 views

Let $\mathbb{R}^\mathbb{Z}$ be topological product, and assume that on $\mathbb{R}$ we have topology $ T_{K} $

Let $\mathbb{R}^\mathbb{Z}$ be topological product, and assume that on $\mathbb{R}$ we have topology $ T_{K} $ where $T_{K} = \{ (a,b) : a,b \in \mathbb{R} , a<b\} \cup \{ (a,b) \backslash K : a,b ...
2
votes
1answer
43 views

Homeomorphism between upper closed hemisphere and disk.

Let $H^2$ be the closed upper hemisphere, that is $$H^2=\big\{(x,y,z\in\mathbb{R}^3)\;|\;x^2+y^2+z^2=1, z\ge0\big\},$$ and let $D^2$ be the closed unit disk $$D^2=\big\{(x,y)\in\mathbb{R}^2)\;|\;x^2+y^...
0
votes
0answers
22 views

Homeomorphism between 2 metric spaces [duplicate]

Is the plane minus four points on the x-axis homeomorphic to the plane minus four points in an arbitrary configuration? In general, how can I show that it doesn't matter which finite points removed ...
0
votes
3answers
57 views

Prove that the product $ X = \Pi_{i \in I} X_{i} $ is discrete $\iff$ $I$ is finite

Let $(X_{i} \colon i \in I ) $ be a family of discrete topological spaces so that $card(X_{i}) \geq 2\ \forall i \in I$ . Prove that the product $ X = \Pi_{i \in I} X_{i} $ is discrete $\iff$ $I$ ...
0
votes
1answer
24 views

Show that nowhere dense set under homeomorphism is nowhere dense

Given homeomorphism $f: X \to Y$ and nowhere dense set $B \subset X$ show that $f(B)$ is nowhere dense set in $Y$. I know that: $f$ is homeomorphism $\implies$ $\left(U \text{ open in } X \iff f(U) \...
0
votes
0answers
22 views

Is there a separable and metrisable topological space $X$ so that $\operatorname{card}(x) > {2}^{\aleph_{0} }$? [duplicate]

Is there a separable and metrisable topological space $X$ so that $ \operatorname{card}(X) > {2}^{\aleph_{0} }$? I can't think of an example.
0
votes
0answers
19 views

The circle S is a topological group, as is the torus T=S*S={(cos2πθ+isin2πθ), cos2πφ+isin2πφ) | 0≤θ, φ≤1} [duplicate]

The circle S is a topological group, as is the torus T=S*S={(cos2πθ+isin2πθ), cos2πφ+isin2πφ) | 0≤θ, φ≤1}. Let H be the subgroup defined by φ=αθ, where α is an irrational number. Show that H is not ...
3
votes
1answer
81 views

A Jordan Curve Symmetric About the Origin Does Not Pass Through Origin

I've been thinking about this statement for a while, and I think it's true, but I'm not sure of how to prove it. The statement is A Jordan curve $J$ that is symmetric about the origin $p$ does not ...
0
votes
1answer
33 views

Vector Space with Hamel Basis is not Separable when all basis elements are 2 apart

Consider a Hamel basis, $\{e_{\lambda}\}_{\lambda \in \Lambda}$, for an infinite dimensional linear vector space. I'm reading something that makes the following claim in passing: Note that if for ...
3
votes
1answer
30 views

Understanding the proof of Kechris' theorem 3.11

In Alexander Kechris' Classical Descriptive Set Theory, he proves a quite useful theorem (3.11) that I'm using as a vital part of a project I'm doing. However, there's a part of the proof I can't wrap ...
1
vote
1answer
19 views

Each convergent filter has at most one cluster point

Let $X$ be a topological space. It is not too hard to show that $X$ is a Hausdorff space if and only if each filter has at most one limit. Suppose now that each convergent filter has at most one ...
-1
votes
1answer
26 views

Is the infinite intersection of nowhere dense sets empty?

Let $X$ be any topological space, and consider a decreasing, countable sequence of nowhere dense sets $A_1 \supset A_2 \supset A_3 \supset \ldots$. Is $\bigcap A_n$ empty? Edit: it seems to be true ...
3
votes
0answers
81 views

Must a subset of $\mathbb{R} ^n$, that has a dimension of $1$, be formed by a union of lines?

Let $S$ be a subset of $\mathbb{R} ^n$ with a box-dimension of $1$. Is $S$ necessarily a union of lines (including "curved" lines and line segments)? If no, then is $S$ at least an infinite set of ...
1
vote
1answer
22 views

Proof Verification: Show that every compact metrizable space has a countable basis

It would be appreciated if someone could review my proof for accuracy. Thanks! Show that every compact metrizable space has a countable basis Proof: Let X be a compact metrizable space. Then ...
2
votes
2answers
34 views

On $ X = (0,1) \times (0,1) $ with anti lexicographic order topology find the character of topological space $X$ ( anti lexicographic order

On $ X = (0,1) \times (0,1) $ with anti lexicographic order topology find the character of topological space $X$ ( anti lexicographic order : $(x_1,y_1) < (x_2,y_2) \iff y_1 < y_2 \text{ or } ( ...
-1
votes
0answers
28 views

Composition operator and topological spaces

I am trying to complete this topological question I have a solution already I just would like for someone to check if for me. Question: My solution:
1
vote
2answers
21 views

Find $\operatorname{Cl }A$ in topological space $\mathbb{R^2}$ with dictionary order topology.

Let $ A = \left\{ (x,y) \in \mathbb{R^2} \mid y= \sin ( \frac{1}{x}) , \ 0 < x \leq 1 \right\}$ . Find $\operatorname{Cl} A$ in topological space $\mathbb{R^2}$ with dictionary order topology. ...
1
vote
0answers
23 views

restriction operators and continuous maps from Hom(X,Y) to Hom(A,Y)

I am trying to solve this topological question and would like to know if I am on the right track with my solution. As in if I have the correct answer or if I need to add or delete anything. However, I ...
2
votes
1answer
32 views

Show that the Compact open topology on Hom(X,Y) is hausdorff

I am trying to complete this topological question and I would like to know if my solution is correct. Any help would be greatly appreciated! My Solution: Let X be a topological space, and Y a ...
1
vote
2answers
41 views

Theorem 24.3 in Munkres's Topology

I'm having a bit of a trouble understanding the proof that if $A$ and $B$ are subspaces of $X$ and $A \subseteq B \subseteq \operatorname{Clo}(A)$, then if $A$ is connected so is $B$ in Munkres's ...
2
votes
1answer
39 views

Complete regularity of infinite dimensional manifolds

A Hausdorff topological space $X$ is called a manifold if $X$ is locally homeomorphic to a locally convex topological vector space. J. Eells, Jr. asserts that every manifold is completely regular (p. ...
0
votes
0answers
21 views

On the invariance of ball $B(a,r)$.

Let $X$ be a complete metric space and a continuous map $ f: X \to X $. A condition sufficient for a ball $ B(a, r) \subseteq X $ to be invariant by $ f $, that is $ f \left (B (a, r) \right) \...
2
votes
1answer
34 views

Better understanding of topology in ://math.stackexchange.com/q/3195705/506847.

Ler $ (\mathbb{N^{*}}, T) $ be topologic space from Prove that there exists a unique topology $T$ for which $ P $ is its subbasis and that topological space $ ( T, \mathbb{N ^{*}})$ is metrisable .. ...
2
votes
1answer
49 views

Prove or find a counterexample: $ Cl(Int(Cl(Int(A))) =Cl(Int(A)) $ $ Int(Cl(Int(Cl(A))) =Int(Cl(A)) $ [duplicate]

Let $X$ be a topological space and $ A \subset X$ a subset of $X$. Prove or find a counterexample: $$ Cl(Int(Cl(Int(A))) =Cl(Int(A)) $$ $$ Int(Cl(Int(Cl(A))) =Int(Cl(A)) $$ I am stuck. Can't find a ...
1
vote
0answers
13 views

Closed points in the closure of a point on a scheme?

Let $X$ be the underlying space of an irreducible scheme. In particular, $X$ is non-empty. Note that the closure of any point is a closed irreducible subset. We say that a point has codimension $n$ ...
1
vote
2answers
33 views

If $U$ and $V$ are disjoint , prove that $ \operatorname{Int} (\operatorname{Cl}U) $ and $\operatorname{Int}(\operatorname{Cl}(V) $ are disjoint sets.

Let $X$ be a topological space and $ U,V \subset X$ open subsets of $X$. If $U$ and $V$ are disjoint , prove that $ \operatorname{Int}(\operatorname{Cl}U) $ and $\operatorname{Int}(\operatorname{Cl}(...
3
votes
1answer
43 views

Prove that there exists a unique topology $T$ for which $ P $ is its subbasis and that topological space $ ( T, \mathbb{N ^{*}})$ is metrisable .

Let $ \infty = \left\{ \mathbb{N} \right\}$ and $ \mathbb{N ^{*}} = \mathbb{N} \cup \left\{ \infty \right\}$. Let $$ P = \left\{ \left\{1\right\} \right\} \cup \left\{ \left\{n, n+1 \right\} \mid n \...
-2
votes
0answers
17 views

product topology and dictionary topology

I'm studying munkres' topology, Sec16 exercise 10. I saw this solution https://dbfin.com/topology/munkres/chapter-2/section-16-the-subspace-topology/problem-10-solution/ but I don't understand the ...
1
vote
2answers
24 views

Prove that topological space $ \mathbb{R^2} $ with dictionary order topology is first countable, but not second countable.

Prove that topological space $ \mathbb{R^2} $ with dictionary order topology is first countable, but not second countable. I am a bit stuck. Some hints would help. For first countability I am having ...
2
votes
1answer
29 views

Check on a proof I saw on another thread: Metrizable Lindelöf spaces have a countable basis

I saw the following proof given of to the theorem below. I don't think the proof is correct, but I wasn't quite sure as it was given an up vote and thought I'd re post here to get some other opinions. ...
0
votes
2answers
25 views

Let $X$ be a topological space and $U$ be aproper dense open subset of $X$. Pick the correct statement from the following

Let $X$ be a topological space and $U$ be a proper dense open subset of $X$. Pick the correct statement from the following (A) If $X$ is connected then $U$ is connected. (B) If $X$ is compact then $...
1
vote
1answer
43 views

For continuously differentiable $f,$ is it true that the set $\{(x_0,x_1)\in (0,1)^2: |f(x_0)| + |f'(x_1)| \geq \epsilon\}$ not compact in $(0,1)^2?$

Notations: We denote $C_0^1(0,1)$ the collection of all real-valued continuously differentiable function $f$ on $(0,1)$ that vanish at boundary, that is, for any $\epsilon>0,$ the set $$\{x\in (0,...
1
vote
1answer
35 views

Property of closed balls and closed sets

Let $(X,d)$ a metric space. Prove that the statements are equivalent : $\textbf{1.}$ For all sequence of closed ball $\{B_n\}$ such that: $B_{n+1} \subseteq B_{n}, \forall n\in \mathbb{N}$ and their ...