Jan 22, 2018 · What, intuitively, does it mean for a structure to be "topological"? I intuitively know what the set of vector spaces have in common, or the set of measure spaces.

For any topological space X, the set of subsets of X with compact closure is a Bornology. If yes to 2, does it coincide with boundedness in a metric space and in a topological vector space? How is it …

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 …

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'.

"A topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space …

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 …

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? …

Dec 25, 2016 · We know about the manifold dimensions of topological objects. I have a query, as every vector space has Hamel basis and the dimension of a vector space is defined as the number of …

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 …

Oct 20, 2016 · I am having some trouble understanding the mechanics of the topological definition of continuity. It has been defined as follows: A function $ f: \\mathbb{R^n} \\to \\mathbb{R^m}$ is …