# Tagged Questions

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### Reference request: Analysis, Algebra and Topology - Same author(s)/publisher(s), progressive order

Is there anywhere I can acquire a collection of all Mathematical undergraduate textbooks by the same publishing author, or authors(so that they are similarly written) and can be completed in a logical ...
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### When is every open set a $F_\sigma$?

My question concerns the $F_\sigma$ property of a topological space $X$. I want to know if there is a particular name for those spaces in which every open set is $F_\sigma$. Moreover, if $X$ is a ...
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### An ultrafilter product topology

Suppose $X=\prod _{i\in\omega}X_i$ is the cartesian product of topological spaces $X_i$ and $u$ is a filter on $\omega$. Define a basis for $X$ by taking the collection of all sets of the form ...
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### Suggestions for a real analysis reference.

Can anyone suggest some real analysis book which has a geometric presentation of the concepts with pictorial representation.
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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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### best intuitive books/video lectures to read topology and functional analysis

What are the best intuitive books/video lectures to read topology and functional analysis ? I am aware of basic linear algebra, analysis and measure theory.
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### Sets that are equal to closure of its interior

Is there a standard name for set $M$ for which $M = \overline{M^0}$? $M^0$ is interior and $\overline{M}$ is closure. Often I work with well behaved sets and functions. I have in mind continuous ...
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### Learning the topology needed for topos theory.

I have just started learning topos theory and I am going through Mac Lane and Moerdijk's book, "Sheaves in Geometry and Logic". I have, unfortunately, very little experience with topology. I started ...
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### Connection between separable measure spaces and $\sigma$-finite measure spaces

I recently came across a theorem which makes a hypothesis that a certain measure space is separable (the definition can be found here). In order to avoid confusion, I'll add the definition here: We ...
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### Raising a partial function to the power of an ordinal

Consider a set $X$, and let $f : X \rightarrow X$ denote a partial function. Then for natural $n$, we can define $f^n$ as iterated composition, e.g. $f^2 = f \circ f$. Now suppose that $X$ is also ...
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### Where to start learning about topological data analysis

I was wondering if anyone could help me out with finding a nice introductory introductory text for topological data analysis (I'm speaking as somebody who has two semesters of experience with ...
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### Approximation by finite sets

I'm reading the book "Topology and Order" by L.Nachbin. In chapter $3$ he speaks about properties of compact Hausdorff spaces. He writes: [A]lthough these spaces may be infinite, they admit ...
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### Proving that $X/R$ is Hausdorff $\implies$ $R$ closed.

As the title says, I'm trying to prove that if $X/R$ is a Hausdorff space then $R\subset X\times X$ is closed. I have several questions about this: $(1)$ What exactly is $R$? I thought of $R$ as an ...
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### Hausdorff dimension mathces Box-counting dimension

I need to compute the Hausdorff dimension of certain sets using a computer and, to date, my approach has been to use a Box-counting algorithm, for I once read that the Hausdorff dimension of an ...
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### Homotopic maps between spheres

I have read somewhere that two maps $f,g:S^n\rightarrow S^n$ satisfying $$|f(x)-g(x)|<2 \qquad \forall \ x\in S^n$$ are homotopic. How can one show this (or does someone have a reference)? I ...
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### The automorphism group of the real line with standard topology

How much is known about the automorphism group of the real line with the standard topology? I have been unable to find a reference for this question. Any information about $\mathrm{Aut}(\mathbb R)$ or ...
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### Good topologies on $\mathcal{P}(X)$

Let $X$ be a topological space, and let $\mathcal{P}(X)$ (resp. $\mathcal{P}_0(X)$) be the set of all subsets of $X$ (resp. the set of all non empty subsets of $X$). Finally, let ...
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### Nagata Smirnov Metrization Theorem

I am looking for a proof for Nagata-Smirnov Metrization Theorem, but I couldn't find one that is readable. I found the paper by Nagata written in 1954 but it is unreadable and uses old notation. ...
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### Good book for general topology [duplicate]

I want a book in general topology with many interesting and hard exercises. I mean a book with topics the same as Munkres but with challenging questions to improve my problem solving ability.
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### How much topology for graph theory?

I am writing a thesis in the context of descriptive complexity in theoretical computer science and therefore need to study a little bit of graph theory. My background is not mathematics but computer ...