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 …

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 …

"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 …

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 …

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 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? …

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.

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.

Jun 13, 2025 · Yes, this simplified version works. In this case, we say that the two metrics are equivalent. To see that it works, notice that it is equivalent to saying that that the identity mapping is …

Sep 18, 2024 · However, regarding the third condition in the definition of a topological manifold, I don't fully understand how $ \varphi $ can be homeomorphic to an open subset of $ \mathbb {R}^n $.