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'.

Apr 1, 2026 · Let $f : X \to Y$ be a continuous function between topological spaces, and suppose $ (x_n) \to x$. I am trying to show that $ (f (x_n)) \to f (x)$ using the following, pretty standard …

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.

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 …

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 …

Jan 4, 2021 · Topology is weird at first, but in the abstract setting of topology you define a topology by saying what your open sets are. This makes it nearly impossible to answer your question, because …

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 …

Aug 5, 2022 · So with respect the last definition I am trying to prove that any topological group $ (X,*,\cal T)$ is completely regular using the following Munkres procedure.

Nov 10, 2025 · Does a dense orbit imply topological transitivity for flows on manifolds? Ask Question Asked 4 months ago Modified 4 months ago

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? …