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? …

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 …

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 …

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 …

"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 ...

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'.

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 …

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 …

Apr 17, 2020 · To expand: the union will be some subset of the original topological space, and, as such, can be given the subspace topology. Munkres's claim is that the union is connected in the subspace …

Oct 20, 2016 · The topological notion of continuity (which is stated for any topological space - even not metric, not only the Rn R n) is a generalisation of the intuitions you may have from the real analysis …