# Topology definition (open/closed)

• October 13th 2012, 02:35 AM
Bamsefar
Topology definition (open/closed)
Is the set from -infinity to +infinity open, closed, neither and/or both?
• October 13th 2012, 02:45 AM
chiro
Re: Topology definition (open/closed)
Hey Bamsefar.

This is an interesting question: I guess one way to look at it is if you can construct a complementary set with respect to the real numbers and the complementary set is the empty set.

So is the empty set open, closed, neither and/or both?
• October 13th 2012, 03:04 AM
Bamsefar
Re: Topology definition (open/closed)
If I'm not mistaken the empty set is both open and closed by definition, which would mean that the answer to my original question is also "both".
• October 13th 2012, 03:19 AM
Plato
Re: Topology definition (open/closed)
Quote:

Originally Posted by Bamsefar
If I'm not mistaken the empty set is both open and closed by definition, which would mean that the answer to my original question is also "both".

In any topology on a set $X$ both $X~\&~\emptyset$ are both open and closed.