Apr 28, 2012 · A topological space is just a set with a topology defined on it. What 'a topology' is is a collection of subsets of your set which you have declared to be 'open'.

While in topological spaces the notion of a neighborhood is just an abstract concept which reflects somehow the properties a "neighborhood" should have, a metric space really have some notion of …

May 23, 2016 · Topological spaces can also be applied to settings where it's not clear how to define a metric, or even when you can't even apply the notion of metric space at all. An important example is …

Feb 12, 2026 · Clearly all cofinal subnets are subnets using the inclusion map, but the converse is false. My query is, if we relax axioms 1-4 above to use cofinal subnets only, what would be the kind of …

Nov 13, 2023 · Combining this with your observation that every subset of a noetherian topological space is quasi-compact, now you know that you only have finitely many pieces of data to keep track of, and …

Thanks for your help! and I'm sorry, what I meant was a sequence in a topological space. I know the examples that you showed me exhibit a non-convergence sequence, but I don't quite see how that …

Dec 6, 2025 · A topological space is locally connected if every point admits a neighbourhood basis consisting of open connected sets. To the definition given by Lee (Introduction to topological …

Jun 18, 2018 · So, the contravariance in the definition of topological continuity is not anything you haven't seen in the metric definition already. You just always thought the metric definition is variant, …

Oct 6, 2020 · Please correct me if I am wrong: We need the general notion of metric spaces in order to cover convergence in $\\mathbb{R}^n$ and other spaces. But why do we need topological spaces? …

Topological dynamics is a subfield of the area of dynamical systems. The main focus is properties of dynamical systems that can be formulated using topological objects.