- #1
center o bass
- 560
- 2
In set theory a set is defined to be a collection of distinct objects (see http://en.wikipedia.org/wiki/Set_(mathematics)), i.e. we must have some way of distinguishing anyone element from a set, from any other element.
Now a topological space is defined as a set X together with a topology T. And one might run into so called 'coarse' topologies (ex. the trivial topology T = {Ø,X}) which are not able to distinguish between the points of the topological space. I.e. suppose we have two points p and q in X. If we had a 'fine topology', so that T contains a lot of sets one could distinguish the points p and q by sets in T to which p belongs and not q. However in the 'coarsest' topologies (as for example the trivial topology above), p and q can not be distinguished in this way.
So my question is; doesn't this contradict the axiom of set theory, that a set must be a collection of element which are distinct in some way? Have I missed some way of distinguishing the point p and q in the example above? Or do we just define the points of a topological to be different elements?
Now a topological space is defined as a set X together with a topology T. And one might run into so called 'coarse' topologies (ex. the trivial topology T = {Ø,X}) which are not able to distinguish between the points of the topological space. I.e. suppose we have two points p and q in X. If we had a 'fine topology', so that T contains a lot of sets one could distinguish the points p and q by sets in T to which p belongs and not q. However in the 'coarsest' topologies (as for example the trivial topology above), p and q can not be distinguished in this way.
So my question is; doesn't this contradict the axiom of set theory, that a set must be a collection of element which are distinct in some way? Have I missed some way of distinguishing the point p and q in the example above? Or do we just define the points of a topological to be different elements?