geometric topology - What is the formal definition of a hole ...
Jan 19, 2024 · To address your title question: There is no formal definition of a hole. The purpose of the whole hole thing is to use our perception of familiar examples (annulus, torus) together with plain …
If $f : X \to Y$ is a continuous function between topological spaces ...
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 …
What exactly is a topological sum? - Mathematics Stack Exchange
Dec 6, 2019 · Why is the topological sum a thing worth considering? There are many possible answers, but one of them is that the topological sum is the coproduct in the category of topological spaces and …
meaning of topology and topological space
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'.
What is the difference between topological and metric spaces?
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 …
Net convergence and relation to topological space
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 …
Where to start learning about topological data analysis?
The linked book seems to be about something else, at least if you believe the description. It seems to be about computational aspects of topology, as opposed to using topological methods for data analysis …
real analysis - What is an open set in a topological space ...
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 …
Difference between the algebraic and topological dual of a topological ...
Sep 11, 2016 · For example, the topological dual (the space of all continuous linear functionals) of a Hilbert space is the Hilbert space itself, by the Riesz representation theorem, while the algebraic dual …
Why do we need topological spaces? - Mathematics Stack Exchange
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? …