16
votes
1answer
464 views

Is Grothendieck classification of tensor norms and Kuratowski's 14 sets theorem somehow related?

It is known that there are only 14 reasonable tensor norms in $Ban$. On the other hand it is well known fact for topologists that one can obtain only 14 different sets from a given set applying ...
2
votes
1answer
133 views

Name of the concept “Topological boundary of A intersected with A”

In closure spaces (thus, also in topological spaces), one may define the boundary of a set A as the closure of A minus the interior of A. This set is partitioned into "the closure of A minus A" and "A ...
3
votes
3answers
286 views

Embedding Theorem for topological spaces, and in general

There are many examples throughout mathematics of abstracting the formal properties of a "familiar" structure, but then having a theorem stating that all models of the abstract axioms embed into one ...
7
votes
3answers
778 views

Importance of separability vs. second-countability

For me second-countability always felt like to be the more important and fundamental concept from general topology than separability. I wonder whether there are any points which can be made for the ...
20
votes
8answers
1k views

Self-containing structures

(This question is partly inspired by What is inter-universal geometry?.) I have absolutely no background in Teichmuller theory or any related subject, but what I can follow of Mochizuki's description ...
8
votes
1answer
658 views

Topology, the board game

Edit: I am reposting this question fom math.stackexchange.com; there may be some professors here who have more experience teaching topology. This is a math education question that I've been thinking ...
3
votes
3answers
525 views

Is there a (standard) name for $\bar{A}\setminus A$?

This is a notation question: If $A$ is a set in a topological space and $\bar{A}$ is its closure, is there a (standard) name for $\bar{A}\setminus A$?
2
votes
0answers
232 views

Descriptive set theory on $\mathbb{R}^\mathbb{N}$

The short version of my question is, What is a good source for learning about descriptive set theory on the space $\mathbb{R}^\mathbb{N}$, under the product topology coming from the discrete topology ...
4
votes
2answers
254 views

Is the generalized Baire space complete?

I want to see whether the fact that the Baire space $\omega^\omega$ is a complete (metrizable) space generalizes to $\kappa^\kappa$ being a complete (topological) space. I think this is an easy ...
1
vote
0answers
114 views

Algebraic properties of the semiring of open subsets.

Does anyone know of a useful general topological application of the algebraic properties of the semiring of open subsets of some topological space? Or examples of any such nontrivial properties at ...
1
vote
1answer
356 views

Is there a countable pseudocharacter Hausdorff spaceļ¼Œsuch that…?

Let X be a Hausdorff space and Difine the Property A as following: if $\mathscr{U}$ is a collection of open sets of X that witnesses Hausdorff property of X (= $\forall x,y \in X$, there exist two ...
1
vote
1answer
297 views

$G_\delta$-diagonal

Could one find a counterexample that a topology space X is Tychonoff, seperable but hasn't a $G_\delta$-diagonal? A topology space has a $G_\delta$-diagonal when there is a sequence ${G_n}$ of ...
6
votes
5answers
710 views

the example of ccc but not separable

I am interested in the relation between the property of countable chain condition (ccc) and the property of separable. Could someone recommend some papers or books about this to me? thanks in advance. ...
5
votes
4answers
776 views

continuous function

Suppose the countable subspace $D$ is dense in the separable Tychonoff space $X$ and $f$ is a continous function from $D$ to the closed unit interval. What are some conditions on $X$ or $D$, which ...
2
votes
1answer
447 views

Meaning of “Compact” in 1932 Paper by van der Waerden “Continuity Theorem for Semisimple Lie Groups”.

I am putting together an exposition on Lie theory; maths research is not my day job, let alone real maths history, so apologies in advance for any ignorance shown by these questions. I am attempting ...
6
votes
6answers
1k views

Elegant representations of graphs in R^3

If I have a graph of a reasonable size (e.g. ~100 nodes, ~40 edges coming out of each node) and I want to represent it in R^3 (i.e. map each node to a point in R^3 and draw a straight line between any ...
17
votes
3answers
1k 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 ...
15
votes
12answers
5k 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: http://www.archive.org/details/introductiontoab031610mbp ...
5
votes
4answers
2k views

Why are inverse images more important than images in mathematics?

Why are inverse images of functions more central to mathematics than the image? I have a sequence of related questions: Why the fixation on continuous maps as opposed to open maps? (Is there an ...
2
votes
2answers
435 views

Conditions useful for proving paracompactness

I have a family of properties which I want to show taken together imply paracompactness (I can show that they are all implied by paracompactness). I can prove a whole bunch of things which are ...
1
vote
2answers
562 views

Existence of convergent subsequences for all values in range?

Consider sequence $s(n) = \sin{nx}$. Are there values of $x$ for which the following holds: For every $y \in \[-1,1\]$ there is a subsequence of $s(n)$ converging to $y$? (Or perhaps just for the open ...
19
votes
7answers
2k views

Why is it useful to classify the vector bundles of a space?

It seems to me that vector bundles are useful because they allow us to bring to bear all of the linear algebra we know to aid in the study of topological spaces. Now, I've read somewhere that it is ...