0
votes
1answer
65 views

What is the metric spaces needed to motivate concepts of general topology?

I intend to start learning some topology on my own. I wonder How much metric spaces I should know in order to motivate the concepts of topology? I know it's possible to learn topology without any ...
0
votes
1answer
22 views

Every metrizable space with a countable dense subset has a countable basis

I'm working on this problem from Munkres: Show that every metrizable space with a countable dense subset has a countable basis. Here's my attempt at a proof. Let $X$ be a metrizable space with ...
0
votes
1answer
39 views

Given a second countable space $X$, show $A \subset X$ has uncountable limit points

Here's my attempt at a solution and I'm wondering if it's correct. Let $X$ have a countable basis with $A \subset X$ an uncountable set. Show $A$ has uncountably many limit points. Let $A'$ be the ...
0
votes
1answer
55 views

Question about $G_\delta$ set

I'm on summer break but I want to keep my math skills sharp so I'm self-studying a bit from Munkres. This question is from pg 194, chapter 4 about the Countability and Separation Axioms. I've ...
1
vote
3answers
52 views

Why is $[0,1]$ not homeomorphic to $[0,1]^2$?

Why is $[0,1]$ not homeomorphic to $[0,1]^2$? It seems that the easiest way to show this is to find some inconsistency between the open set structures of the two. It is clear that the two share the ...
0
votes
1answer
19 views

Covering of a universal space and covering of a subset

It seems to me that the definition of covering depends on the set we are referring to. For example, if $X$ is the universal topological space, then a collection $\mathcal A$ of subsets of $X$ is said ...
3
votes
3answers
145 views

How to show a collection of sets is countable

In the proof for "Every open subset $\mathcal O$ of $\mathbb R$ can be written uniquely as a countable union of disjoint open intervals" Stein and Shakarchi (2005 p6) argue that (after having defined ...
1
vote
2answers
53 views

Uniqueness of Topology and Basis

In measure theory, we know there is a (unique) minimal $\sigma$-algebra generated by a generator. I am wondering whether this applies to topology and its basis. There are two directions to consider ...
0
votes
0answers
42 views

How to deform a curve in specific manner

I am wondering whether we can deform a path in specific ways continuously i mean if there is a closed piece wise $C^1$ smooth path which has to be deformed to another piece wise $C^1$ smooth path. Let ...
2
votes
1answer
76 views

Doubts in definition of continuity in a topological space

EDITED Let $(X, \mathscr T_X)$ and $(Y, \mathscr T_Y)$ be two topological spaces. Let there be a function $f: X \to Y$. Then $f$ is said to be continuous if for every $E \in \mathscr T_Y$ the ...
0
votes
1answer
29 views

understanding Continuity in topological space

Let $(X, \mathscr T_X)$ and $(Y, \mathscr T_Y)$ be topological spaces and $f: X \to Y$. $f$ is continuous iff $f^{-1} (E) \in \mathscr T_X $ for every $E \in \mathscr T_Y$. My doubt is: I dont know ...
2
votes
3answers
53 views

Understanding the definition of compactness.

$(X, \mathscr T )$ be a topological space and $A \subset X$. $\{ U_i \mid i \in I \}$ is said to be an open cover of $A$ if $A \subset \cup_{i \in I} U_i$. $A$ is said to be compact if there exists ...
0
votes
0answers
37 views

X is a compact topological space?

Let $X$ be a topological space. Then, when it is said that $K \subset X$ is compact, it is clear to me that every open cover of $K$ contains finite subcover of $K$. But what do we mean when it is ...
0
votes
1answer
33 views

triangle inequality on a given metric

$X$ be set consisting of all sequences $(x_1,x_2, \dots)$ s.t $x_i \in \mathbb R$ and $\sum x_i^2$ converges I need to prove triangle inequality for the metric on $X$ given by, $d(x,y) = [ ...
2
votes
1answer
53 views

If a continuous function$f: (X, \mathscr T_X) \to (Y, \mathscr T_Y)$ is injective (Given $Y$ is Hausdorff), show that X is hausdorff

$(X, \mathscr T_X)$ and $(Y, \mathscr T_Y)$ be 2 topological spaces. $Y$ be Hausdorff. $f$ be a continuous function, $f: X \to Y$. To show that if $f$ is injective $\implies$ $X$ is Hausdorff. ...
0
votes
3answers
77 views

Counter Example about Continuous Functions

(James Munkres page 104 Theorem 18.1) Let $X$ and $Y$ be topological spaces; let $f: X \rightarrow Y$. If $f$ is continuous, then for every subset $A$ of $X$, one has $f(\overline{A})\subset ...
1
vote
1answer
53 views

Sub-Basis of a Topology James Munkres

James Munkres defines a subbasis $\mathcal S$ for a topology on a set $X$ as a collection of subsets of $X$ whose union equals $X$. Then the topology generated by the subbasis $\mathcal S$ is defined ...
3
votes
1answer
120 views

Help proving Cantor Intersection Theorem using Bolzano-Weierstrass Theorem

Came across the following exercise in Bartle's Elements of Real Analysis and am quite unsure about my solution. Would greatly appreciate it if someone could take a look at it. The Bolzano - ...
1
vote
1answer
54 views

How is an open set defined without referring to any distance function in a topology?

I am currently studying general topology. The definition given by Royden looks very confusing to me. It says that the elements of a topology is called open sets without actually defining what exactly ...
2
votes
2answers
189 views

$A \times B$ is an open set in $\Bbb R^2 \implies A$ and $B$ are both open in $\Bbb R$; $A,B \neq \emptyset$

I am studying Analysis on my own. Reading The Elements of Real Analysis by Bartle. Came across the above problem and I came up with the following solution but am very unsure about it. Would be very ...
2
votes
2answers
82 views

Example for a set in $\Bbb R^p$ whose interior is $\emptyset$ and closure is $\Bbb R^p$

The following exercise was in the Elements of Real Analysis by Bartle. Give an example of a set $A$ in $\Bbb R^p$ such that $A^{\circ} = \emptyset$ and $A ^ - = \Bbb R^p$. Can such a set be ...
0
votes
2answers
41 views

Regular or normal topological space

How do we define an open neighborhood around a closed set? My question is with respect to a normal or regular topological space where we use the concept of an open neighborhood around a closed set.
2
votes
0answers
77 views

Difference between Metric Space and Topological Space

I am reading Chapter 11 of Real Analysis written by Royden and Patrick (4th). It says "The concept of uniform convergence of a sequence of functions is a metric concept. The concept of pointwise ...
3
votes
1answer
167 views

A subset of $\Bbb R^p$ is open iff it is the union of a countable collection of open balls

I am studying analysis on my own and need some help verifying the solution to the above exercise found in Bartle's Elements of Real Analysis. I know there are other posts answering the same question ...
0
votes
0answers
53 views

Generating an open set using basis of standard topology.

Let $(\mathbb R, \mathscr T$) be a topological space where $\mathscr T$ be a standard topology. Let $K = \{ \frac1n | \; n \in \mathbb N \}$. How can I generate $K$ from the basis elements of this ...
0
votes
0answers
34 views

Confusion in an order topology being Hausdorff and T1.

There is a theorem which says that every order topology is Hausdorff. Also every Hausdorff follows $T_1$ Axiom. So suppose $X = \{1,2\}$. Now $ 1<2$. The bases for the topological space $X$ can be ...
0
votes
0answers
45 views

Proof verification and suggestion to elude the AC (equivalent definition of adherent points).

Hi everyone I'd like to know if the following is correct and, more importantly, if there is some way to escape of the axiom of choice (as the hint the book says "use AC"). Definition: Let $X\subset ...
1
vote
2answers
88 views

Elementary properties of closure

Hi everyone I'd like to know if the following is correct. I really appreciate any suggestion. (Honestly the only one that matters me is the second property the others are easy, I think) Thanks. ...
0
votes
1answer
147 views

Verification and help to simplify an argument about closure of some sets.

Hi everyone I'd like to know if what I have so far is correct, I think is much work for something which is too simple I would appreciate any advice or whatever. Moreover, I have doubt in (3) and (4), ...
3
votes
3answers
100 views

Is the concept of a uniform space elementary?

I'm self studying with Munkres's topology and he uses the uniform metric several times throughout the text. When I looked in Wikipedia I found that there's this concept of a uniform space. I'd like ...
0
votes
1answer
107 views

countably compact subset of a metric space is sequentially compact?

I'm currently trying to get my head around a proof in Lipschutz [1]. Solved exercise 21 of Chapter 11 (page 164) asks for a proof of the following statement: Let $A$ be a countably compact subset of ...
0
votes
1answer
103 views

A basis question of least upper bound property.

A set $A$ is said to have least upper bound property if every subset $A_0 \subset A$ has a least upper bound. $\mathbb{R}$ has least upper bound property is well known. Now consider the subset $A = ...
1
vote
1answer
73 views

Cantor's nested interval theorem is satisfied on every linear continua?

Cantor's nested interval theorem is satisfied on the real line in usual topology. Linear continuum is a generalization of the real line for a topological space. Will this theorem be satisfied on any ...
2
votes
1answer
242 views

Continuous function between real line and unit circle

Let $f : S^1 \rightarrow \mathbb{R}$ be a continuous map. Show that there exists a point x in $S^1$ s.t. $f(x) = f(-x)$. Thank you.
6
votes
4answers
446 views

Can a function with just one point in its domain be continuous?

For example if my function is $f:\{1\}\longrightarrow \mathbb{R}$ such that $f(1)=1$. I have the next context: 1) According to the definition given in Spivak's book and also in wikipedia, since ...
1
vote
3answers
2k views

Proving Every open set in $\Bbb R$ is a countable union of open intervals. [duplicate]

This question is from William R. Wade's Introduction to Analysis book: Prove that every open set in $\Bbb R$ is a countable union of open intervals. I have no ideas honestly. Thank you.
2
votes
3answers
114 views

Prove that $a$ is a cluster point of $E$ $\iff$ for each $r>0$, $E\cap B_r(a)$ \ $\{a\}$ is nonempty.

Question: Let $E$ be a subset of $\Bbb R^n$ Prove that $a$ is a cluster point of $E$ $\iff$ for each $r>0$, $E\cap B_r(a)$ \ $\{a\}$ is nonempty. definiton: A point $a \in \Bbb R^n$ is ...
3
votes
1answer
160 views

Prove that a proper subset $E$ of $\Bbb R^n$ is connected $\iff$ it contains exactly two relatively clopen sets.

Prove that a proper subset $E$ of $\Bbb R^n$ is connected $\iff$ it contains exactly two relatively clopen sets. I researched the meaning of "clopen set". And I reached the result that so as to ...
3
votes
1answer
118 views

What is $\mathfrak a$?

I'm currently reading Mendelson's Introduction to topology and have came across this theorem: Theorem 3.8: Let a neighborhood in a topological space be defined by Definition 2.2 and an open set in ...
2
votes
1answer
61 views

When in topology is $A = f^{-1} \circ f[A]$ or $B = f \circ f^{-1}[B]$ true, for an $f$ which is not one-to-one?

I'm having a bit of trouble with an example problem in the topology book I'm reading. It's problem #11 (pp 104) of the "Solved Problems" section of Chapter 7, of the Schaum's Outline for "General ...
1
vote
4answers
81 views

Measure nonzero implies dense on a rectangle

This would be a very handy lemma for me but I have been unable to prove it thus far. If $S \in \mathbb{R}^n$ is bounded and is not of measure zero, then there exists a rectangle $R$ such that $S$ ...
2
votes
3answers
118 views

Lebesgue covering theorem

I am having trouble understanding Lebesgue covering theorem as stated in Mathematical Encyclopedia. First of all I think I have confusion with the definition of "finite subsystem". Is it finite ...
2
votes
3answers
1k views

Resources for self-study of general topology

Could you point me to good resources for self-study of general topology. I want to learn the basics and how to prove theorems about structures like polyhedra by myself.
5
votes
1answer
394 views

Want to learn differential geometry and Riemannian geometry by myself

I want to follow a Ph.D. programme on geometric analysis. The main focus are Monge-Ampere equations and convex level set of $p$-harmonic equations. The theory of PDEs is very familar to me. However, I ...
4
votes
2answers
335 views

How much Set Theory before Topology?

I was reading Baby Rudin for Real Analysis and wanted to explore Topology a little deeper. I bought George Simmons' Introduction to Topology and Modern Analysis and found myself liking it. I am having ...
3
votes
2answers
301 views

study topology: homotopy and homology

I want to study the basis of topology. I know functional analysis and very basic topology. I need to learn about homologies and homotopies but it seems that all the books (mostly of Russian authors) ...
2
votes
1answer
251 views

Group acting on locally finite tree

Let $G$ be a group acting on a locally finite connected tree $T$ i.e. each vertex degree is finite. Let $G$ has compact open topology i.e. for each compact set $C$ and an open set $U$ of $T$, the sets ...