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)
Example of a vector space that is not a topological space? A topological space is not a vector space because well, it's just not It doesn't satisfy the things it's required to satisfy in order for it to be a vector space I give you for example the so called Sierpinski Space This is clearly not a vector space A vector space is in turn not a topological space unless you define a topology on it
Topology of a circle - Mathematics Stack Exchange Topological properties are properties which are preserved under homeomorphisms Continuity forbids you to cut things, but to transform them by stretching Notice here that the definition for continuity totally depends on the topologies $\mathcal{O}_X$ and $\mathcal{O}_Y$, and therefore on open sets!