All Questions
Tagged with big-list gn.general-topology
21 questions
333
votes
34
answers
96k
views
Why is a topology made up of 'open' sets? [closed]
I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of ...
67
votes
22
answers
10k
views
When has discrete understanding preceded continuous?
From my limited perspective, it appears that the understanding
of a mathematical phenomenon has usually been achieved,
historically, in a continuous setting
before it was fully explored in a discrete ...
67
votes
10
answers
12k
views
Non-homeomorphic spaces that have continuous bijections between them
What are nice examples of topological spaces $X$ and $Y$ such that $X$ and $Y$ are not homeomorphic but there do exist continuous bijections $f: X \to Y$ and $g: Y \to X$?
48
votes
19
answers
17k
views
What is your favorite proof of Tychonoff's Theorem?
Here is mine. It's taken from page 11 of "An Introduction To Abstract Harmonic Analysis", 1953, by Loomis:
https://archive.org/details/introductiontoab031610mbp
https://ia800309.us.archive....
37
votes
14
answers
5k
views
What are interesting families of subsets of a given set?
Motivation
The usual starting point of both Topology and Measure Theory is the definition of a family of subsets of a set $S$.
Indeed, one defines a topology on $S$ to be a family of subsets ...
- 33.3k
37
votes
5
answers
5k
views
Locales and Topology.
As someone more used to point-set topology, who is unfamiliar with the inner workings of lattice theory, I am looking to learn about the localic interpretation of topology, of which I only have a ...
31
votes
17
answers
14k
views
Applications of Brouwer's fixed point theorem
I'm presenting Brouwer's fixed point theorem to an audience that knows some point-set topology. Does anyone have any zippy / enlightening / cool applications or consequences of it? So far, I have:
...
31
votes
13
answers
6k
views
Classic applications of Baire category theorem
I've seen Baire category theorem used to prove existence of objects with certain properties. But it seems there is another class of interesting applications of Baire category theorem that I have yet ...
30
votes
8
answers
3k
views
Cryptomorphisms
I am curious to collect examples of equivalent axiomatizations of mathematical structures. The two examples that I have in mind are
Topological Spaces. These can be defined in terms of open sets, ...
24
votes
15
answers
5k
views
Applications of connectedness
In an «advanced calculus» course, I am talking tomorrow about connectedness (in the context of metric spaces, including notably the real line).
What are nice examples of applications of the idea of ...
18
votes
8
answers
2k
views
Concepts in topology successfully transferred to graph theory and combinatorics with non-trivial applications?
What are some of the difficult concepts in topology that have been transferred to graph theory and combinatorics where a certain new application has been found.
A good example is Lovász's proof of ...
16
votes
12
answers
5k
views
Examples of $G_\delta$ sets
Recall that a subset $A$ of a metric space $X$ is a $G_\delta$ subset if it can be written as a countable intersection of open sets. This notion is related to the Baire category theorem. Here are ...
- 18.7k
15
votes
5
answers
2k
views
Striking existence theorems with mild conditions, and simple to state: more recent examples?
I would like to write an article about powerful existence theorems that assert, under mild and simple conditions, that some basic pattern or regularity exist. See some examples below. By mild ...
15
votes
3
answers
1k
views
Why it is convenient to be cartesian closed for a category of spaces?
In 1967 Steenrod wrote what later became a quite celebrated paper, A convenient category of topological spaces (Michigan Math. J. 14 (1967) 133–152). The paper conveys the work of many (among the most ...
- 9,111
13
votes
5
answers
1k
views
Connectedness in the plane
There are several open problems in topology which concern connectedness and subsets of the plane. The biggest of these is undoubtedly:
Question. Does every non-separating plane continuum have the ...
- 955
8
votes
1
answer
716
views
Topological fraction rings and fields
Linked to this question
and as a sequel to my answer of it.
Let $R$ be a topological (commutative, unital) ring and set $S$ be a submonoid of $(R,\times,1_R)$.
Let
$$
s_{frac}\ :\ R\times S\to S^{-...
- 3,738
6
votes
14
answers
5k
views
Applications of compactness [closed]
Similar to this question: Applications of connectedness I want to collect applications of compactness.
E.g.: compact + discrete => finite, which can be used to prove the finiteness of the ...
3
votes
2
answers
326
views
Examples of TVS with no non-trivial open convex subsets
I give here the classical example of the space $E = L^p([0,1])$ which has no open convex subsets apart from $\emptyset$ and $E$. Consequently, there is no non-trivial continuous linear form on $E$.
...
- 1,355
2
votes
3
answers
530
views
examples of lifting properties
A number of seemingly unrelated elementary notions can be defined uniformly with help of (iterated) Quillen lifting property
(a category-theoretic construction I define below) "starting" to a single (...
- 41
2
votes
1
answer
384
views
Properties of the weak-$*$ topology
Let $X$ be a topological affine space over a complete base field $\mathbb S := \mathbb C$, $\mathbb R$ or $\mathbb Q_p$. Let $X^*$ be the dual space of continuous affine functionals equipped with the ...
1
vote
1
answer
908
views
What are the topological properties of the metric space retained (inherited) for its completion
Let $(X,d)$ be a metric space and $(\bar{X},\bar{d})$ its completion.
There is a list of topological properties
Wikipedia - Topological property
Does anybody know list which of them are retained (...