Understanding the Concept of Open and Closed Sets in Topology

[Ka Yan]

What's the difference between those assertions:
" A set X is both open and closed."
and
" A set X is neither open nor closed."

For the first, I knew some examples: The real line itself, and the empty set.
But what example can be araised about the second?
And any better ones to the former?

Thx.

 

[morphism]

What's the difference between those assertions:
" A set X is both open and closed."
and
" A set X is neither open nor closed."

Those two statements are complete opposites!

For the first, I knew some examples: The real line itself, and the empty set.
But what example can be araised about the second?
And any better ones to the former?

If you're working strictly in the real line with its usual topology, there are no other examples of sets that are both open and closed. Can you try to prove this? And as for sets that are neither open nor closed, what can you say about something like [0,1)?

On the other hand, if you work with arbitrary topological spaces, then the situation is different. For example, in any discrete space, every set is both open and closed.

 

[HallsofIvy]

Actually, it is much easier to find examples of sets, in the real line, that are neither open nor closed, than both open and closed. In the real line with the "usual" topology, the only sets that are both open and closed are the empty set and R itself while, as morphism said, any "half open" interval, [a, b) or (a, b], is neither open nor closed.