How is the Void Open in the Topology of R^n?

[quasar987]

I'm not asking why it is an axiom of a topology that the void is open, but rather, when the topology of R^n is developped and open sets are defined as sets such that for any point in the set, we can find an epsilon-ball centered on that point that is entirely contained in the set.

My book says that it follows from the dfn that the void is empty. How is that? If we argue that "since the void has no point, then it is true that for all points, we can find and epsilon-ball, etc.", then the opposite is just as true: "Since there are no point, we can say that for all point, we can never find an epsilon-ball, etc."

There is no points in the void, so the definition simply does no apply it seems!Similar question: what's the boundary of the void? is it the void or the whole of R^n?

 

[DeadWolfe]

My book says that it follows from the dfn that the void is empty. How is that? If we argue that "since the void has no point, then it is true that for all points, we can find and epsilon-ball, etc.", then the opposite is just as true: "Since there are no point, we can say that for all point, we can never find an epsilon-ball, etc."

You're correct that both arguments are valid, but note that "Every point does not have X" is NOT the opposite of "Every point has X". In order for the null set not to be open, we would need that the contradiction of "Every point has X" hold, which is of course "there is a point which does not have X".

There is no points in the void, so the definition simply does no apply it seems!

The definition always applies. Either every point has a property, or one of the points doesn't.

Similar question: what's the boundary of the void? is it the void or the whole of R^n?

Well the null set is closed, so that ought to answer your question.

 

[quasar987]

Ok, I see! Thx DW

 

[DeadWolfe]

No problem. It does sound like you might want to brush up more on your logic if you intend to study mathematics at this level.

 

[quasar987]

This has been my conclusion as well

 

[mathwonk]

consider a statement of form "for all elements of set S, property P is true".

If S is empty this statement is true, :"vacuously".

this aNSWERS YOUR QUESTION, say in a metric space. i.e. openness is defined by a ":universal" quntifier: "for all p in S, there is an open ball around p also contained in S".

 

[fopc]

consider a statement of form "for all elements of set S, property P is true".

If S is empty this statement is true, :"vacuously".

this aNSWERS YOUR QUESTION, say in a metric space. i.e. openness is defined by a ":universal" quntifier: "for all p in S, there is an open ball around p also contained in S".

S contains all its limit points.
If rephrased (i.e., as a universally quantified implication) and taken as definition of closed set in a metric space, we arrive at same vacuous truth (i.e, antecedent false) wrt. empty set.

Empty set both open and closed.