Feb 6, 2026 · What is the relation between the four variants? The Wikipedia variant is a stronger requirement than Hatcher's. Locally simply connected spaces are semilocally simply connected and …

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 …

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 …

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? …

"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 …

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'.

Oct 15, 2023 · My book uses the abbrevation $\text {Top}^2$. I want to make sure I don't confuse it with the category $\text {Top} \times \text {Top}$. The topology $\text {Top}^2$ has arbitrary pairs of …

Jun 13, 2025 · At the end, what I want to show is that for topological spaces with the metric topology, τ(B) = τ(B) if and only if for each x ∈ X and for each ϵ> 0, there exist real numbers δ1, δ2> 0 …

Feb 1, 2026 · 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.

Apr 16, 2014 · 23 To a topologist, topological groups are interesting in their own right. The group structure actually gives us interesting topological structure, too!