0
votes
0answers
21 views

Textbooks to complete concurrently - Self learning empowerment

A user is completing some year challenge that takes them through $9$ textbooks and they are alternating in author. Algebra - Cohn Analysis - Rudin Topology - Lee Repeat three times. I would like ...
1
vote
0answers
9 views

Fundamental domain for a $C_2$-action on a Stone space

The following result seems to be true (I can prove it, only quite indirectly): Let $X$ be a Stone space (i.e. a compact totally disconnected Hausdorff space) and $\sigma : X \to X$ be a ...
3
votes
1answer
72 views

Reference Request to Prepare for Hatcher's “Algebraic Topology”

Hatcher himself has an excellent and always generously free set of notes on point- set topology: http://www.math.cornell.edu/~hatcher/Top/TopNotes.pdf It includes up to quotient spaces. It seems ...
2
votes
1answer
185 views

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 ...
2
votes
0answers
44 views

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 ...
3
votes
1answer
65 views

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 ...
1
vote
3answers
120 views

Suggestions for a real analysis reference.

Can anyone suggest some real analysis book which has a geometric presentation of the concepts with pictorial representation.
2
votes
1answer
74 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 ...
6
votes
1answer
167 views

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.
1
vote
0answers
42 views

Extending a homeomorphism between subspaces

Lavrentiev's Theorem. Suppose $X$ and $Y$ are complete metric spaces, $A\subseteq X$, $B\subseteq Y$, and $f:A\to B$ a homeomorphism. Then $f$ can be extended to a homeomorphism $\overline f :G\to ...
0
votes
1answer
55 views

Where can I read about the topological properties of the perforated plane?

Anybody knows about perforated plane in topology? What is it? Where can I read about it? I'm talking about the plane $\mathbb{R}^2$ with the topology that have basis elements disks without finite ...
4
votes
1answer
36 views

Topological property: set-theoretically large subsets of an infinite space are not compact.

Let $X$ be an infinite topological space. Say that $X$ satisfies # if no subset of $X$ of cardinality $|X|$ is compact. So for instance it is clear that no (infinite) compact space satisfies # any ...
1
vote
1answer
81 views

On the density of $\mathcal{C}^\infty(\Omega) \cap W^{1,\infty}(\Omega)$ in $W^{1,\infty}(\Omega)$

Let $\Omega$ be an open set of $\mathbb{R}^n$ ($n\ge 1$). We know that the Meyers-Serrin theorem isn't true in $W^{1,\infty}(\Omega)$. But is it true that $\mathcal{C}^\infty(\Omega) \cap ...
3
votes
1answer
65 views

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 ...
3
votes
3answers
133 views

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 ...
5
votes
1answer
152 views

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 ...
5
votes
0answers
62 views

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 ...
5
votes
2answers
161 views

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 ...
1
vote
1answer
33 views

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 ...
2
votes
2answers
87 views

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 ...
0
votes
1answer
66 views

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 ...
-1
votes
1answer
33 views

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 ...
10
votes
4answers
308 views

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 ...
2
votes
0answers
46 views

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 ...
1
vote
0answers
62 views

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. ...
2
votes
1answer
83 views

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.
3
votes
2answers
45 views

Non-empty intersection between a compact and an unbounded connected subset of $\mathbb{R}^d$

I am quoting from MathOverflow, where I have just read it as part of a comment: "If $C$ and $S$ are, resp., a compact and a connected unbounded subset of $\mathbb{R}^d$ such that $C\cap S \ne ...
2
votes
2answers
28 views

looking for Theorem 3.22 of Cardinal functions in topology

read an article which uses Theorem 3.22 of "Cardinal functions in topology, ten years later". I searched this book on internet but is not there. I searched it in my university library but isn't there ...
0
votes
1answer
105 views

Reference request: Partition of unity…

I was looking for some material that could help me understand a real analysis course (1st year undergraduate). My teacher treated the following topics: Partition of unity Existence of regular ...
8
votes
3answers
303 views

Categorical introduction to Algebra and Topology

I am self-studying Mathematics in my free time. At the moment I am reading books on Algebra and on Category theory. More exactly, I started working through the book $\textit{Algebra}$ by Serge Lang. I ...
3
votes
0answers
46 views

A stronger condition than planar graph?

Is there a name for this condition on a graph: a graph that can be embedded in the plane (planar), in such a way that of its univalent vertices do not lie inside any face? So, one can think of this ...
0
votes
0answers
18 views

Projective topology (reference)

Please suggest me a good reference to study Projective topologies. I just want an introductory exposition.
6
votes
2answers
157 views

Prerequisites for Freedman's proof of the 4-dimensional Poincaré conjecture

I have a good understanding of differential geometry, enough at least to understand many details of Hamilton & Perelman's approach to the 3-dimensional Poincaré conjecture. I have no such ...
0
votes
0answers
79 views

Totally bounded uniform spaces vs proximity spaces (need proof)

nLab says "The category of totally bounded uniform spaces and uniformly continuous functions is equivalent to the category of proximity spaces and proximally continuous functions". How to prove this? ...
0
votes
1answer
36 views

Cauchy filters defined for proximity spaces?

I define in my draft article Cauchy filters $\mathcal{X}$ on a uniform space $\nu$ by the formula: $$\mathcal{X}\ne\bot \wedge \mathcal{X}\times^{\mathsf{RLD}}\mathcal{X}\sqsubseteq\nu.$$ ...
3
votes
1answer
162 views

Are continuous mathematical models of discrete physical phenomena messy because of a disconnect between “continuous” and “discontinuous”?

Examples from statistical mechanics and continuum mechanics abound: a discrete phenomenon (e.g. kinetic energy of molecules) is "averaged" out over the constituents of the system to which it applied ...
2
votes
1answer
38 views

“Topologification” of a subcollection of a power set

Let $X$ be any set and consider any $\mathscr{S} \subseteq \mathcal{P}(X)$, where the latter is the power set. It is natural to ask if we make $X$ a topological space by the subcollection ...
3
votes
0answers
86 views

Applications of the Kuratowski closure-complement theorem

I crossed with the Kuratowski closure-complement theorem while learning Munkres's Topology (Problem 21 in Section 17; Page 102, 2nd edition). The following description is from B.J. Gardner and M. ...
2
votes
3answers
91 views

An example of a non first countable Fréchet-Urysohn space?

As the head title says, I need a Fréchet-Urysohn space but not first countable, (on the way, a good Text book to follow). Thanks.
5
votes
2answers
145 views

Good source for a point set topological introduction to CW complexes?

Most algebraic topology books I found don't dwell too much on point set topology of CW complexes. I'd like too become more familiar with them. Anyone knows a good source (with exercises) too learn ...
2
votes
1answer
61 views

Projective limit and connected components

Let $E$ be a topological space. Let $\mathcal{K}$ be the set of the compact subsets of $E$. $(E-K)_{K \in \mathcal{K}}$ is a projective system, because if $K,K'$ are two compacts, there are two ...
6
votes
0answers
152 views

Point set topology from an algebraic perspective?

I got this idea of viewing a topology as an operation on a ring of sets. Let $\mathcal R = (\mathcal P(X), \cap, \triangle)$ be a ring of sets. ($\triangle$ is the symmetric difference operation and ...
0
votes
2answers
35 views

Properties of the topology with the basis consisting with the sets $A_n = \{n, n+1, \dots \}$, $n \in \mathbb{N}$.

Let $X = \mathbb{N}$ be equiped with the topology generated by the basis consisting of the sets $A_n = \{n, n+1, \dots \}$, $n \in \mathbb{N}$. Is $X$ compact? connected? Hausdorff? My Attempt : The ...
3
votes
0answers
106 views

Comparision of Textbooks for Topology

I will take Introduction to Topology in next semester. The professor lists two reference books, Topology (by Sheldon Davis)/Topology (by Munkres). I know the textbook written by Munkres is a classic ...
3
votes
0answers
73 views

References for the Čech-Stone compactification of Hyper-Reals?

It seems like $\beta\mathbb R$ has been heavily studied, but I am interested in learning more about $\beta(\mathbb R ^\omega /u)$. $\mathbb R ^\omega /u$ is a proper extension of $\mathbb R$ when ...
4
votes
1answer
105 views

Contracting a contractible set in $\mathbb R^2$

Assume that $A$ is compact, connected and contractible set in $\mathbb{R}^{2}$ (for example: simple square). If we contract this set to a point the space still will be homeomorphic to ...
5
votes
3answers
205 views

Topology Exercises Books

I am taking next semester a class on Topology and was wondering if anyone could advise me a book containing a lot of exercises WITH solutions in order to train. Thank you in advance
1
vote
1answer
65 views

Name this topological space

For each $n \in \mathbb{N}$, define the circle $S_n$, which passes through the origin by: $$S_n := \left\{ (x,y) \in \mathbb{R}^2 : x^2+\left(y-\tfrac{1}{n}\right)^2=\tfrac{1}{n^2}\right\}$$ I know ...
3
votes
0answers
51 views

Relative continuity of a function

I've come across the following generalization of continuity in a rather surprising place. Let $X$, $Y$, and $Z$ be topological spaces and $f:X\to Y$ and $g:Y\to Z$ be functions. Say $f$ is continuous ...
6
votes
2answers
83 views

Has the idea of generalizing the codomain of a metric been seriously considered?

The long line is much longer than $\mathbb{R}$, and indeed many chains have this property. Thus, since metrics are usually assumed to be real-valued, this can be understood as an assumption that ...