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 …

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 …

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 …

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 …

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.

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 …

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 …

Dec 19, 2020 · For once, there is a nice continuity of notation. However, there are multiple topological notions of dimension: topological dimension, both inductive dimensions, dimension of CW …