Topology Definition and 818 Threads

My question comes from homework from a section on the Tychonoff Theorem. This is the question: Now I have an idea about how to go about this. I know that Q is compact since I = [0,1] is and the Tychonoff Theorem states that the product of compact spaces is compact. I then know that f(Q)...

I'm trying to prove some stuff that involves the projection map, say p:X x Y ->X. But I need to know if it's continuous. If a map is continuous, then the preimage of a open/closed set is open/closed. The problem is, what do open sets in X x Y look like? I know what the basis elements are...

:bugeye: Does anybody have any suggestions on how to prove that a topological space X is COUNTABLY compact (i.e. every COUNTABLE open cover has a finite subcover), IF AND ONLY IF, EVERY NESTED SEQUENCE of closed nonempty subsets of X has a nonempty intersection? I also need hints on how to...

Hi, I am typing up my topology homework and I want to make the cool looking t I see in the book. How do I accomplish this? Thanks!

I'm really just having trouble figuring out what a question is asking. Here's the question: My problem is really just that in proving that say regularity is a local property, I'm not sure what to use as a subspace. I could take a given base set and then consider the rest of the base sets...

I can not figure out what is wrong with this : suppose one deals with a multiple connected universe, such as a torus. In order to make it simple, let us imagine we consider two very massive objects in this topology, say two well separated clusters of galaxies whose distances are large compared...

How useful is topology for physics? And what are soome good books for learning topology. I find a lot of the definitions in textbooks way too abstract and not giving examples of the topological spaces they are defining. Drop some titles if you have a moment.

What exactly is topology? I know it's used a lot in modern physics, but what other applications does it have? Now, a little bit on the theoretical side, what's difference between point-set, algebraic, geometric and differential topology? Can anyone provide an example problem on each? What...

If S is a set with the discrete topology and f:S->T is any transformation of S into a topologized set T, then f is continuous. Can someone help me prove this? I have no idea where to even begin.

Hi, I am not sure if this would be the right place to post this but i know that it is a mathematical concept. I have read a bit about topology in the latest scientific american, and it really intrigued me. I am fascinated by this idea. Therefore i ask if you would kindly point me in the...

can somebody tell me how to unlink 2 linked tori using continuous deformations only? Also are there any free software tools for visualizing topological operations?

I have an idea of what topology is but I am clueless as to what applications it has? Anybody have any idea what topology is used for?

In a quantum gravity discussion ("Chunkymorphism" thread) some issues of basic topology and measure theory came up. Might be fun to have a thread for such discussions. for instance the statement was made, apparently concerning the real line (or perhaps more generally) that a countable set...

Hello everybody, I am facing a Topology problem, and I hope you may be able to help me. Let me try to describe my problem as clearly as I can: assume you have a set F of functions, such that any element f in F is a one-dimensional bounded and continuous function with common support S...

Let f be a real-valued function defined and continuous on the set of real numbers R. Which of the following must be true of the set S = {f(c): 0<c<1}? I. S is a connected subset of R II. S is an open subset of R III. S is a bounded subset of R The answer is I and III only. I understand...

a line segment (including its endpoints) is toplogically equivalent to a point. consider S a nonempty totally ordered (ie a set with a relation <= such that <= is reflexive, transitive, x<=y and y<=x imply x=y, and every pair is comparable--i think that's what total order means anyway) set...

Greetings, Having decided on a physics/math double major next year, I decided to get a head start this summer. After tackling some classical mechanics, my next target is topology. My problem is the following: I have been informed that there are two approaches to the subject, one...

What are the main differences in approach between standard? topology and algebraic topology?