Topological Definition and 265 Threads

I'm reading this book (Hartshorne), and it uses a funny definition of topological dimension, which I'm having a hard time convincing myself is the usual one. The definition is as follows: dim X is the supremum of natural numbers such that there exists a chain Z_0\subset Z_1\subset \dotsb...

eddo's thread got me thinking: How can you tell if a specific topological space is compact? It seems like it would be hard to do just starting with the definition of compactness.

How can I show that F:X\times I\to I given by F(x,t)=(1-t)f(x)+tg(x) is continuous, given that f:X\to I and g:X\to I are continuous (here I is the unit interval [0,1]). It seems that F is continuous, but I want to show that explicitly. Any help appreciated! X is any topological space. (I...

http://lanl.arxiv.org/pdf/quant-ph/0501041 the pioneer anomaly is due to topological phase defect of light

I don't know if this paper was commented on here while I was away at Christmas time, but I was just pointed to it by a paper on today's arxiv and I think it desrves notice. http://arxiv.org/abs/hep-th/0411073 Robert Dijkgraaf, Sergei Gukov, Andrew Neitzke, and Cumrun Vafa; Toplogical...

The critical density determines the Universe evolution along time. So, by measuring this density, we could know about the finite or infinite age of the Universe. But we don't know from General Relativity if our Universe is spatially finite or infinite. As far as I know, such question...

This is a chess/math puzzle I invented: Consider a fully set up chess board (in starting position). Invent a condition (new rule) in which black wins without either side making any moves. Note: this problem has a really abstract and *topological* approach. :smile: Please email me...

I was wondering if knotted topological strings can be asigned definite quantum charges apart from mass and angular mumentum. That would really be useful to glueball people who try to find candidate in the hadronic spectrum : they use only mass and angular momentum. If they had a mean to assign...

Is it true that a perfect generalized ordered space can be embedded in a perfect linearly ordered space? It is true that a perfect generalized ordered space can be embedded as a closed subset in a perfect linearly ordered space.

What is topological defects in cosmology? Is this a current field of research and what are their progresses?

I have printed a notes about differential geometry, and it says: -A Coo differentiable structure on a locally Euclidean, Hausdorff topological space M of dimension m is a collection of coordinate systems F Then it specifies the conditions that F must satisfy, but I'm a little lazy and won't...

I'm a noob starting out studying differential geometry and topology. Really probably somewhere in the multivariate calculus level, but I've been trying to understand the plethora of terminology I'm encountering with this higher math. I've been reading a lot on Wikipedia.org and PlanetMath.org...

I believe that the Axioms for TQFT were set out by Atiyah in 1990 and that one of the equivalent definitions of a TQFT is in category terms: a TQFT is a functor from the category of n-dimensional cobordisms to the category of Hilbert spaces, satisfying certain conditions. Is anyone familiar...

From the abstract: "An action principle is described which unifies general relativity and topological field theory. An additional degree of freedom is introduced, and depending on the value it takes the theory has solutions that reduce it to (1) general relativity in the Palatini form, (2)...

Chern-Simons theory is a 3d-TQFT and Crane-Yetter model is a 4d-TQFT There exist another Topological Quantum Field Theory apart of these two models?