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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1answer
32 views

Graph of Topologist's Sine Curve

I'm looking for whether the graph of topologist's sine curve and closed topologist's sine curve are closed or not. But due to some misconception, I'm facing problems with this. $\underline{\text{...
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0answers
23 views

Property of a covering

My problem: Suppose $(X,d)$ is a metric space and $A=\bigcup_{i \in I}B(x_i,r_i) \subset X$ is totally bounded where $B(x,r)=\{y \in X: d(x,y)<r \}$. I know that if $A$ is totally bounded, $A \...
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0answers
17 views

If $T$ is a continuous bijection on $\mathbb R^d$, and $M$ is a $C^1$-submanifold with boundary, is $T(\partial M)=\partial T(M)$?

Let $d\in\mathbb N$ and $T:\mathbb R^d\to\mathbb R^d$ be bijective and continuous. I'm not sure how to approach this, but if $M\subseteq\mathbb R^d$, does $T$ map the topological boundary/interior of $...
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1answer
37 views

Does $\mathbb{Q}$ have the finite-closed topology?

Let $\mathbb{Q}$ be the set of all rational numbers with the usual topology Does $\mathbb{Q}$ have the finite-closed topology? My attempt : I think yes Finite - closed topology mean cofinite topology ...
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2answers
33 views

Rudin proof for compact subsets $\{ K_{\alpha} \}$ (theorem 2.36) — Contrapositive or contradiction?

I am having doubts about Theorem 2.36 pasted below. I was able to follow all the steps individually, but I don’t see how this is a proof by contradiction. It seems to be it is a proof by ...
3
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0answers
28 views

Show that a certain space of banded matrices with nonnegative determinants is connected and closed

Let $n$ and $k$ be two positive integers, with $n \geq 2$ and $k \geq 1$. Let $l=\{{l_i}\}$ be an $nk$-dimensional positive real vector. Let $A_{n,k}(l)={(a_{i,j})}_{1 \leq i,j \leq nk}$ be the ...
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1answer
41 views

How could a path be homotopic to a point

I have been following Serge Lang's Intoduction to Complex Analysis at a Graduate Level and I met this theorem. I want to ask what does it mean for a function to be homotopic to a point? I am only ...
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0answers
18 views

Relationship between the co-finite topology and standard topology on R? [closed]

Relationship between the co-finite and left ray and countable and standard topology on R?
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0answers
25 views

Base point of BU/BO, classifying space of U/O.

There are principal bundles $$U(n) \to V_k(\mathbb{C}^n) \to G_k(\mathbb{C}^n)$$ and $$O(n) \to V_k(\mathbb{R}^n) \to G_k(\mathbb{R}^n),$$ where $V_k(\mathbb{F}^n)$ and $G_k(\mathbb{F}^n)$ are the ...
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1answer
37 views

Lee's Intro to Topology, generating the same topology

Suppose $M$ is a set and $d, d^\prime$ are two different metrics on $M$. Prove that $d$, and $d'$ generate the same topology on $M$ if and only if the following condition is satisfied: for every $x \...
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0answers
46 views

Closure / neighborhood system in Zariski topology

*How can every neighborhood of a limit point contain a closed set $v$ when $v$ has a complementary set in those neighborhoods ? *What is a good example of a Zariski closed dense set ? So far I have ...
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5answers
61 views

Show that $E = [0,1]$ is not open.

Here is my proof. Not sure if I'm doing it correctly. We recall that $E$ is open provided every point of $E$ is an interior point. We then have $\forall p \in E$, there exists a neighborhood of $p$, $...
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1answer
61 views

Prove that $G \cong \pi_{1} (X/G)$.

This is a question from an exam. Let $X$ be a topological space which is simply-connected, and let $G$ be a group of homeomorphisms of $X$ which acts properly discontinuously, meaning $$\forall \ x\in ...
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2answers
82 views

If sets $A$,$B$ are closed, convex sets, they have the same boundary and their interiors intersection is non-empty, does $A=B$?

If sets $A$,$B$ in Euclidean space are closed, convex sets, they have the same boundary and their interiors intersection is non-empty, can we say $A=B$? Any suggestions and comments are welcome!
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1answer
50 views

'Fake' identity regarding the closure in the subspace topology

I have the following argument which I encountered, and can't seem to find why it's not true: Let $X$ be a topological space and let $A$ and $B$ be nonempty subsets of $X$. Then $\overline{A\cap B}^A=\...
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1answer
30 views

Union of continuous images of a compact set in $\mathbb{R}^n$

The continuous image of a compact set, $L \subseteq X$, is compact. But how about if I have a set of continuous operators $F \subseteq A$ where $A = \{ \text{continuous operators } T:L \rightarrow X\}$...
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2answers
31 views

Proving that if $\forall n\in\mathbb N,\exists x_n \in \mathbb R: |x_n - a| < \frac{1}{n}$, then $a \in \bar S$

This exercise is in my general topology textbook: Let $S$ be a non-empty subset of $\mathbb R$ and $a \in \mathbb R$. Prove that $a \in \bar S$ if and only if $\forall n\in\mathbb N,\exists x_n \in S:...
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2answers
88 views

If sets $A, B$ in Euclidean space are closed sets, they have the same boundary and their interior's intersection is non-empty, can we say $A=B$?

If sets $A, B$ in Euclidean space are closed sets, they have the same boundary and their interiors intersection is non-empty, can we say $A=B$? Any suggestions and comments are welcome!
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1answer
12 views

Variant of Quotient Metric is an Ultrametric

Let $(X,d)$ be a metric space and define an equivalence relation $\sim$ on $X$. Then $$ d'([x],[y]):= \inf\{d(x',y'): x' \in [x],\, y' \in [y]\}, $$ may fail the triangle inequality, where $[x]$ is ...
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2answers
58 views

Properties need to define Derivatives on Topological space

I just started learning topology and was curious about defining derivatives on general topological spaces. Since we can define continuous functions on Topological spaces, my question is what ...
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1answer
22 views

Convergent sequence and Hausdorff space

It's trivial that any sequence in a Hausdorff space converges to at most one point. What about its inverse? E.g. If a topological space have the property that any sequence in it converges to at most ...
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2answers
56 views

Norm Inequality: Other

I need help proving this inequality to understand a preliminary remark. $$\Bigg |\Bigg| \dfrac{x}{\|x\|}-\dfrac{y}{\|y\|}\Bigg |\Bigg| \geq \|x-y\| $$ with $x$ and $y$ satisying $\|x\|,\|y\|\leq 1$ ...
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0answers
37 views

Continuous bijection between surface of sphere and $[0,1] \times [0,1]$ [closed]

Is there continuous bijection between surface of sphere and $[0,1] \times [0,1]$? spherical coordinate system is not bijection. so i want to know that there exist continuous bijection. I think there ...
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0answers
34 views

Specific book recommendations for topology and analysis [duplicate]

I'm looking for books on real and complex analysis (including measure theory), and topology (including algebraic topology) that are written in a style as close as possible to Dummit and Foote's ...
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1answer
33 views

Topologically complete and $G_\delta$ theorem proof

I don't understand the underlined part of the proof. It is very important, but it is not obvious. Since ($S$, $e$) is complete, $y\in S$.
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0answers
31 views

Topology of Distribution via limit of topological vector space

As far as I know, we can define the topology of distribution (either on T or on noncompact nonempty set $\Omega$ ). I read one note and find we can also see the topology as one type of limit. Can ...
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0answers
21 views

Proof of Smoothness of Projection Maps & Maps to Cartesian Product

Loring Tu's in his An Introduction to Manifolds discusses the smoothness of a projection map in Example 6.17, and the smoothness of a map to a Cartesian product in Exercise 6.18 (Second Edition, page ...
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0answers
17 views

Show that strong deformation retract is preserved under adjunction [closed]

If $A$ is a strong deformation retract of topological spaces $X$ and $Y$, show that for each continuous map $F:A \to Y$ the subspace $Y$ is a strong deformation retract of adjunction space of $X$ and $...
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1answer
11 views

Mapping one tree onto another

Let $T$ and $T'$ be trees (finite acyclic graphs in the obvious compact connected topology). Let $f:T\to T'$ be a continuous surjection. Let $A'$ be an arc in $T'$ (by arc I mean $A'\simeq [0,1]$). ...
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1answer
50 views

Set of isolated points closed? [duplicate]

$\varnothing\neq A \in 2^\mathbb R$, $A^s$ is isolated points of $A$. Is $A^s$ closed set? notation: x is isolated point means $\exists r>0 [N(x;r)\cap A=\{x\}]$ I would like to prove but I have ...
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1answer
41 views

confused about quotienting $\Bbb R^2$ by $\Bbb Z^2$ vs. compactifying $ \Bbb R^2$ first and then gluing sides

Learning a little about quotient spaces and I don't understand something. (1) Compactify $\Bbb R^2$ to $[0,1]^2$ then glue sides to make torus. (linked post gives example of compactification) (2) $\...
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2answers
22 views

Is a collection of open sets is a basis for a topology on $X$ if it gives a basis for a dense subset of $X$?

Let $Y$ be a dense subset of a topological space $X$. Let $\mathcal B := \{U_\alpha : \alpha \in \Lambda\}$ be a collection of open subsets of $X$ such that $\{U_\alpha\cap Y : \alpha \in \Lambda\}$ ...
5
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3answers
92 views

Bijective map from a set to a subset of reals?

There is a concept that I have been thinking about quite a lot lately as I am currently self-studying point-set topology: Say we have a bijective map from one interval, $[a,b]$, to another interval, $[...
1
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1answer
26 views

Continuous function on a compact domain

I have a simple question about continuity: I know that continuous functions send compact sets to compact sets, but I am confused with a concrete example. Consider $X = \{\frac{1}{n}: n \in \mathbb{N} \...
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1answer
29 views

Baire's Theorem with locally compact Hausdorff space

Statement of the theorem: If $S$ is either a) a complete metric space, or b) a locally compact Hausdorff space, Then the intersection of every countable collection of dense open subsets of $S$ is ...
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0answers
60 views

"Can we find a continuous bijective function from $(0, 1)$ to $(5, 6) \cup (7, 8)$? [closed]

Can we find a continuous bijective function from $(0, 1)$ to $(5, 6) \cup (7, 8)$? Kindly help me out, please. Here as we can see that the image set is disconnected then by using the contrapositive ...
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0answers
44 views

Is there a straightforward group homeomorphism between the 2-adic integers and $[0,1)$?

I think the (set of the) ring of 2-adic integers is represented by: $$\left\{\sum_{i=0}^\infty 2^ix_i:x_i\in\{0,1\}\right\}$$ And I think the set of real numbers in the interval $[0,1)$ is represented ...
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1answer
48 views

curve and topology

Let $ \sigma : \mathbb{R} \to \mathbb{R}^2$ that maps t to $(\frac{t}{1+t^4}, \frac{t}{1+t^2})$. Show that this curve is regular and injective, but it is not homeomorphic to $Im( \sigma )$. I've found ...
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1answer
20 views

Finite Product of path-connected spaces

Problem Finite product of path connected spaces is path connected. Attempt Suppose $X_1$ and $X_2$ be path connected spaces. Now, I am using Maps into Products. Let $X_1×X_2$ isn't path connected. ...
2
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3answers
59 views

Homeomorphism between $Y$ and $\{b\}\times Y$

If I have two topological spaces $(X,\mathcal{T}_x)$ and $(Y,\mathcal{T}_y)$. I am trying to show that the space $\{b\}\times Y$ is homeomorphic to Y, where $b\in X$ It is my understanding that if I ...
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1answer
26 views

Are T1 and T0 inheritable in the subspace and product topologies?

T2-ness property is inherited in subspaces and in the topological product. What about T1 and T0, are they inheritable? Can someone give simple examples when this happens and when this does not happen, ...
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3answers
43 views

Convergence and Comparison of Topology

Let $(X,\mathcal{T}_1)$ and $(X,\mathcal{T}_2)$ be a topological space endowed with two different topologies. If any convergent net $\{x_v\}$ in $(X,\mathcal{T}_1)$ is convergent in $(X,\mathcal{T}_2)$...
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1answer
21 views

Give a function uniformly continuous with respect to one metric and not with respect to another, while both induce the same topology

I would very much appreciate an example to the above question or some hints to construct one. Such a function should not exist in normed vector spaces: If the topologies induced by two norms are equal ...
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2answers
49 views

About the one point compactification [closed]

I have a question about the one-point compactification, given an arbitrary topological space $(X,\tau)$, does the one-point compactification always exist? Thanks!
2
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1answer
31 views

A simple homeomorphism between unit circle and quotiented unit circle

I want to show $S^1/\mathscr{R}$ and $S^1$ are homeomorphic where $S^1$ is the unit circle and the equivalence relation is $$(x',y')\mathscr{R}(x'',y'') \iff y''\leq 0 \text{ and } y'\leq 0.$$ Now I ...
0
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1answer
17 views

Clarification on the use of subsequences to prove that in a metric space a sequence in a compact subset admits a convergent subsequence in the subset

Lemma: Let $(X,\tau)$ be a topological space and let $K \subseteq X$ compact. If $E\subseteq K$ is infinite then $Der(E)\neq \emptyset$ , where $Der(E)$ is the set of accumulation points Theorem Let (...
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1answer
65 views

Covering the closed interval $[0,1]$ with open intervals around rational points? [closed]

For each rational point $x \in [0,1] \cap \mathbb{Q}$ let $\epsilon_x > 0$. Does $$ [0,1] \subseteq \bigcup_{x \in [0,1] \cap \mathbb{Q}} (x - \epsilon_x, x + \epsilon_x ) $$ always hold? Thank you
1
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1answer
36 views

Finding limit points of sets in $(\mathbb Z , \tau)$

In my general topology textbook there is the following exercise: Let $(\mathbb Z , \tau)$ be the set of integers with the finite-closed topology. List the set of limit points of the following sets: 1 ...
2
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2answers
31 views

Why do we require $K$ to be compact instead of just finite in $(X=\mathbb{R} \cup \{P\}, \tau_2=\tau_e \cup \{X\setminus K\})$ for compactness?

Example 1: Let $\tau_{disc}$ be the discrete topology We know $(\mathbb{R},\tau_{disc})$ is not compact If we add a point P so that we have $X=\mathbb{R} \cup \{P\}$ and define a new topology as $\...
2
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1answer
60 views

Prove continuous maps $f,g: S^{n} \rightarrow S^{n}$ are homotopic if $f(x)\cdot g(x)=0$ for all $x\in S^n$.

Let $f: S^{n} \rightarrow S^{n}$ and $g: S^{n} \rightarrow S^{n}$ be two continuous map on $S^{n}$. Suppose that, for all $x \in S^{n}$, we have $$f(x)\cdot g(x)=0\,.$$ Then show that $f \simeq g$. ...

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