# Tagged Questions

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### Topology of a nested sequence of subsets

Hi everyone I'd like to know if the following proof is correct, I think so. And also if there is a more direct approach without the many subcases. Thanks in advance Let $X$ be an infinite set, and ...
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### Baker's transformation: continuity, orbits of irrational and rational points

I've reading the Pugh's Analysis book and I have problems with one exercise. This says: The baker's transformation: a rectangle of dough is stretched to twice its length and folded back on itself. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 - ...
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### 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 ...
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### $A \times B$ 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $S$ ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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 ...