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.

Filter by
Sorted by
Tagged with
0
votes
0answers
8 views

Separability between $X$ and $\beta X$

Q) Is it true that if $\beta X$ is separable, then $X$ is also separable? I know if $X$ is locally compact, the result is true. Can someone help me with this?
0
votes
0answers
19 views

What's the difference between 1- Trace Topology, 2- Subspace Topology, 3- Relative Topology, and 4- Induced Topology?

I need help with some topology terms... So what is the difference between: 1- Trace Topology 2- Subspace Topology 3- Relative Topology 4- Induced Topology Or are they all the same thing?
1
vote
4answers
42 views

A closed set is intersection of countable collection of open sets

I am currently reading Apostol Mathematical Analysis. There was this question Question : Prove that a closed set in $\mathbb{R}^1$ is the intersection of countable collection of open sets. [Here $N_a(\...
0
votes
0answers
25 views

Question about proof of closure of set.

Let $\Gamma$ be the Moore Plane Topology and consider the following set: \begin{equation*} A=\bigcup_{q\in\mathbb{Q}\setminus \{ 0\}} \left(\{q\} \times \left[0,\frac{1}{d_q}\right]\right). \end{...
0
votes
0answers
23 views

Boundedness of a set in a Metric space.

In the book 'Topology and modern analysis'by Simmoms, page 69 the problem 3 says: "Show that a subset of a metric space is bounded iff it is non-empty and is contained in some CLOSED sphere"....
3
votes
1answer
43 views

One-point compactification of $\Bbb{R}$: is $-\infty=\infty$?

On $\Bbb{R}$, there are separate notions of divergence to $-\infty$ and $\infty$. However, any continuous function has a maximum and a minimum on $\Bbb{R}^*$ since it is compact. If the function in ...
0
votes
0answers
14 views

Computing $H_1(K)$ and $H_2(K)$, where $K$ is the complex consisting of the proper faces of a 3-simplex

I am interested in computing $H_1(K)$ and $H_2(K)$, where $K$ is the complex consisting of the proper faces of a 3-simplex. However, specifically, I am interested in doing so, if possible, by using ...
1
vote
2answers
33 views

Opening a cube does not change its homotopy type

Consider the (skeleton of a) cube. I want to compute the fundamental group. Now I know that if you "open the cube" - like you would open a box - or in another pov you splat it onto a plane, ...
0
votes
0answers
38 views

Fundamental group of a bucket

I want to compute the fundamental group of a bucket. Wrong proof: If I retract the bucket onto the disk at the base what I get is a disk with a handle attached to it by two points, which in turn is ...
0
votes
1answer
27 views

Prove all continuous maps from $X$ to $Y$ are constant maps when $X$ is endowed with a trivial topology and $Y$ is a Hausdorff topological space.

Let $X$ be a set endowed with the trivial topology. Let $Y$ be a Hausdorff topological space. Show that all continuous maps from $X$ to $Y$ are constant maps. I need help with the above ...
1
vote
1answer
23 views

Does the axes of $\mathbb R^n$ have the fixed-point property?

I was studying Croom's Principles of Topology and was asked to decide whether the set $$A=\{(x_1,x_2)\in\mathbb R^2:x_1=0\text{ or }x_2=0\}$$ has the fixed-point property. I first thought about ...
0
votes
1answer
22 views

Metrizability of product spaces, Horst Herrlich, Topology I

In Horst Herrlich's Topology I, page 117, Statement 4.4.12 the author goes on to prove that product space is metrizable if each component ($\underline X_i$) of the product space is metrizable and the ...
0
votes
0answers
14 views

Show that in box topology the set $\mathbb{R}^\mathbb{N}$ of real sequences is connected.

My idea is think in 2 subsets of $\mathbb{R}^\mathbb{N}$ the real limited sequences and not limited. With this show that $\mathbb{R}^\mathbb{N}$=$\mathbb{R}^\mathbb{N}\cup \emptyset$. But i don't know ...
0
votes
1answer
22 views

Homeomorphism betwen Unit circle in the Taxicab Geometry to the Unit circle in the Euclidean Geometry

I need make a homeomorphism to the unit circle in the Taxicab Geometry to the Unit circle in the Euclidean Geometry, but I can't find information about a parametrization to the unit circle in the ...
2
votes
2answers
22 views

Difference between topologies generated by a basis and a subbasis

I am looking for examples where topologies generated by a subbasis and a basis yield to the same topologies, preferably in a finite topological space. For instance let $X=\{1,2,3\}$. The collection $\...
0
votes
1answer
27 views

Showing a subset is open with respect to a topology

I have asked and read similar questions, but I am still somewhat confused on the notion of "a set being open with respect to some topology". My task is concerning the box topology and the ...
0
votes
1answer
45 views

Fundamental group of a bouquet of circles

Consider a bouquet of $n$ circles, centered at $(1,0), (2,0),...,(n,0)$ $$ W_n= \bigcup_1^nC_r$$ with $$C_r:(x-r)^2+y^2=r^2$$ I want to compute the fundamental group of this space. $W_n$ is path ...
1
vote
1answer
16 views

Two questions of first and second category sets in complete metric spaces

On the proof of If $X$ is a complete metric space, then any dense $G_\delta$ in $X$ is residual in $X$. I'm confused on how to prove this statement. $G_\delta = \cap_{n=1}^\infty U_n$ where every $...
0
votes
1answer
28 views

Equivalence relation on the disjoint union of all sections of a presheaf - verify transitivity

Let $X$ be a topological space and associate to each open subset $U \subset X$ a set $S(U)$ in such a way that whenever $V \subset U$ is another open subset the so called restriction maps $\rho_V^U \...
0
votes
1answer
17 views

Basic question on local compactness

Let $X$ be a topological space. Let $A \subset F \subset X$. Let $F$ be a subspace of $X$. Let $A$ be a subspace of $F$. Suppose that under this subspace topology inherited from $F$, $A$ forms a ...
0
votes
0answers
11 views

Not-continuos map with its graph closed.

Find an example of a map $f:(X,d_{X})\rightarrow (Y,d_{Y})$ between metric spaces such that $X$ is compact, $f$ is not continuous but $\Gamma(f):=\lbrace (x,f(x))\in X\times Y\mid x\in X\rbrace$ is ...
1
vote
0answers
34 views

Every continuous map $f:\mathbb{C}P(2) \to \mathbb{C}P(2)$ has a fixed point, without Lefschetz theorem.

I would like to know if there is a nice proof of the fact that every continuous map $f:\mathbb{C}P(2) \to \mathbb{C}P(2)$ has a fixed point, without use of the Lefschetz fixed point theorem.
0
votes
3answers
44 views

Show if the following sets are open [closed]

I'm doing some problems and there are some to which I simply don't have an answer. Are the following sets open or closed? $(0,1)\text{ embedded in }\mathbb R^2\colon\{(x,y)\mid x\in(0,1),y=0\}\...
0
votes
0answers
33 views

Convergence of $f \circ g_n$ towards $f \circ g$

Let $f\in C^1([0,1]\times\mathbb{R},\mathbb{R})$, and $g_n \in C^1([0,1],\mathbb{R})$ that converges uniformly towards $g \in C^1([0,1],\mathbb{R})$ : i.e $$||g_n-g ||_{\infty}\rightarrow0. \,\, (n\...
0
votes
0answers
20 views

Relation between open balls in extended real line and real line

Consider the metric defined on $\overline{\mathbb{R}}$ using the usual bijection $g:y\mapsto \frac{y}{|y|+1}$. Take $0<r$ and $x\in\mathbb{R}$. Then, what is the relation between $B_{\mathbb{R}}(x,...
3
votes
3answers
65 views

Why is the unit circle not homeomorphic to the closed unit disk?

I know that the unit circle = $\{(x,y): x^2+y^2 =1\}$ is not homeomorphic to the closed unit disk = $\{(x,y): x^2+y^2 \leq 1\}$, but I'm not sure how to prove it. I've tried with arguments with cut-...
0
votes
0answers
47 views

Computing $\mathrm{Vect}_k(M)$?

$\mathrm{Vect}_k(M)$ is the isomorphism classes of real $k$ rank vector bundles over $M$. In Bott-Tu book they give only an example: For contractible manifolds it is $\mathrm{Vect}_k(M)$ is a point. I ...
1
vote
2answers
17 views

Open neighborhood of a subset of a metric space

Suppose $X$ is a metric space, with distance function $d:X\times X\to \Bbb R$. Also suppose $U$ is an open neighborhood of a subset $A\subset X$. Then for each $a\in A$, we can choose an $\epsilon_a&...
4
votes
3answers
74 views

Metrizable Topological Space In Many Ways

Prove that if a topological space $(X, T)$ is metrizable then it is metrizable in infinitely many ways. $$$$As the given topological space is metrizable so there exists a metric $d$ on the set $X$ ...
1
vote
2answers
20 views

Definition of a Basis for a Topology - Intersection of Basis Elements and Possibility of being a Topology

Part of the definition of a basis for a topology for a set $X$ states: If $x$ belongs to the intersection of two basis elements $B_1$ and $B_2$, then there is a basis element $B_3$ containing $x$ such ...
0
votes
0answers
12 views

If $\mathcal C\subseteq C_b(E)$ is an algebra, $fe^{-\varepsilon f^2}$ can be approximated by functions from $\mathcal C$

Let $E$ be a complete separable metric space, $\mathcal C\subseteq C_b(E)$ be a point-separating$^1$ algebra$^2$, $\mu_i$ be a probability measure on $(E,\mathcal B(E))$ and $\varepsilon>0$. We ...
1
vote
3answers
31 views

Continuous map between subsets of topological spaces

I know very little about topology, so this is a rather basic question. A continuous map between topological spaces $X$ and $Y$ is defined as a function $f\colon X\to Y$ such that the preimage of any ...
0
votes
1answer
25 views

Complete Set-Cauchy Sequence

Consider $(\Re ^{2},\left \| \cdot \right \|)$ with $\left \| x_{1}^{2}, x_{2}^{2} \right \| = \sqrt{x_{1}^{2} + x_{2}^{2}}$. Is the set $\{(x_{1} ,x_{2})\in \Re ^{2} \mid x_{1}\neq 0 \text{ and } x_{...
15
votes
1answer
497 views

Does a Lie group's group structure (not Lie group structure) determine its topology?

Does a Lie group's group structure (not Lie group structure) determine its topology? Said another way, can you have two Lie groups that are isomorphic as groups but not homeomorphic? If so, the group ...
0
votes
1answer
30 views

If $G\subset \mathbb{R}$ is an additive group such that $\inf{G^+}=0$, then $G$ is dense in $\mathbb{R}$

Suppose $G$ an additive group of real numbers, that is $x,y\in G\implies x-y\in G$ with $G\neq \{0\}$. Consider $G^+$ the set of positive elements of $G$. If $G\subset \mathbb{R}$ is an additive group ...
0
votes
2answers
22 views

Show that the plane with countably many points removed is path-connected under the Euclidean topology. [duplicate]

Show that the plane with countably many points removed is path-connected under the Euclidean topology. This is quite an interesting issue, hopefully everyone will help you
0
votes
1answer
26 views

Limit of addition mapping at a point $(a,\infty)$ w.r.t. $\mathbb{R}\times\mathbb{R}$ is $\infty$

With respect to $\mathbb{R}\times\mathbb{R}$, the function $(x,y)\mapsto x+y$ has a limit at every point $(a,b)$ of $\overline{\mathbb{R}}\times\overline{\mathbb{R}}$, except at the points $(-\infty,\...
0
votes
1answer
43 views

How can I find the limit points of this set?

I have $a_n = (-1)^n +\frac{2}{n}, \quad n=1,2,3,... \quad$ and $\quad A=${$a_n : n = 1,2,3...$} How can I find the limit points of this set? And also how do I know if this set has any isolated ...
0
votes
1answer
36 views

A question of connectedness of subspace

My question is about Lemma 23.1 in James Munkres' topology. Suppose $X$ is a topology and $Y\subset X$. Suppose we found two (specific) sets $A$ and $B$ that are disjoint, non-empty, and their union ...
2
votes
1answer
50 views

A topological manifold whose boundary is $S^1 \lor S^1$

I need to find if it exists a topological manifold (with boundary) whose boundary is $S^1 \lor S^1.$ I think there isn't any. Indeed, any topological manifold whose boundary were $S^1 \lor S^1,$ the ...
1
vote
0answers
44 views

Counting “cardinality” of bounded subsets

Imagine that I have a bounded subset $X \subset \mathbb{R}$. $X = (0,1) \cup (2,3) \cup (4,5) \cup \{9\}$ I want to count the number of intervals + singleton sets that make up $X$. To be clear, ...
2
votes
0answers
28 views

Find $X$ such that $\partial X = S^1 \times T$

Let $Y= S^1 \times T$ where $T$ is the torus. I need to define a manifold with boundary $X$ such that $\partial X = Y.$ I know that in general if $A$ is a manifold with boundary and $C$ is a manifold ...
1
vote
1answer
29 views

Prove that a curve is an homeomorphism

Check if the parametrization of the curve $\gamma :\mathbb{R}\rightarrow\mathbb{R}^{2}$ defined by $$\gamma(t):=(\frac{t}{1+t^{4}},\frac{t}{1+t^{2}})$$ is an homeomorphism on $Im(\gamma)$ This map ...
1
vote
0answers
36 views

locally compact

Definition 1: An open subset $U$ in topological space in $ (X, \tau) $ is regular open if $ \alpha(U)= \operatorname{int}(\operatorname{cl}(U)) = U $. Since the collection of regular open sets in a ...
1
vote
0answers
39 views

Some equivalence relations on $S^1 \times S^2$

Let $S^1: x^2+y^2=1$ be the circle ; $S^2: x^2+y^2+z^2=1$ be the sphere; then define also $X= S^1 \times S^2$ I want to define an equivalence relation on $X$ such that $X/\sim$ is not a top manifold. ...
-1
votes
2answers
46 views

Proof - Product Topology. [closed]

Show that \begin{align*} (\mathbb{R}^n, \mathcal T_{eucl}) \times (\mathbb R^m, \mathcal{T}_{eucl}) = (\mathbb{R}^{n+m}, \mathcal T_{eucl}). \end{align*} Any tips on how to get started? Thanks.
0
votes
1answer
19 views

Definition Of Open Subsets Of Metric Spaces

I want to know that if there is a metric space $(X, d)$ and an open subset $A \subset X$ and if we change the metric from $d$ to say $d'$, then is it possible that $A$ will be no longer an open subset ...
0
votes
0answers
13 views

Is an isolated zero also a regular value

Is an isolated zero also a regular value? The definition of a isolated zero is: A point $p_0 \in M$ is called an isolated zero of $X$ if $X(p_0) = 0$ and there exists an open set $U \subset M$ such ...
0
votes
1answer
26 views

Showing a set is open with respect to box vs product topology

I am currently studying topology with Munkres and I am asking for some general proof techniques and clarification rather than just posting my question and wait for a solution. I am given that the real ...
1
vote
0answers
18 views

Examining an upper semi-continuous function on the empty topology

Let me start out with the question that prompts this: Original Problem: Let $X$ be a compact topological space, and let $f : X \to \Bbb R$ be an upper semi-continuous function. Show that $f$ attains ...

1
2 3 4 5
908