general topology - Boundedness in a topological space? - Mathematics . . . To define boundedness on topological vector spaces, you're using the extra structure: either the semi-norms used to define the topology, or in general the scalar product The point I was making is that a bornology is a way to abstract the notion of boundedness which is available in some contexts (metric spaces, top vector spaces)
general topology - Definition of a topological property - Mathematics . . . "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 X possesses that property every space homeomorphic to X possesses that property
meaning of topology and topological space 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' But declaring a set to be 'open' isn't quite enough: we want our open sets to be 'nice' in some way, and we want to be able to perform set operations on them to preserve this niceness
Why do we need topological spaces? - Mathematics Stack Exchange Intuitively, a topological space is all the data you can collect about the points comprising an object using a collection of rulers, while ignoring the sizes of the rulers - that last part is the essence of the whole famous old joke that a topologist can't tell the difference between hir coffee mug and hir doughnut, at least before taking a
What is the difference between a manifold and a topological manifold? Another confusion might arise as all second level structures give raise to the non-examples, in that logic, non-topological manifolds (e g What is a non-topological manifold (if such a thing exists)?) In short, all topological manifolds are manifolds and all manifolds are defined only with a topological manifold
What is a topological space good for? - Mathematics Stack Exchange The thing about a topological space is that you can take finite intersection and still remain in the topological space For instance, with $\tau_1$, when you are interested about $\{b,c\}\cap\{a,b\}$, you don't get an open set, which isn't practical This doesn't happen in a topological space like $\tau_2$
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 "nearness" and hence, the term neighborhood somehow reflects the intuition a bit more