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kinda yes!!
http://en.wikipedia.org/wiki/Open_set#Open_and_closed_are_not_mutually_exclusive

Are you okay with why it's definitely not open?

i am guessing it to be neither open nor closed set.

But in R2, you need a circle, not just an interval:
|dw:1356810632181:dw|

Circle and Sphere are actually misleading. I should have said Disk and Ball.

According to the definition on Wikipedia, this can't be clopen either.

That's right. A clopen set is both open and closed, this set is neither open nor closed.

Let's see what other people have to say ... i'll leave the post open

Any idea on Cantor intersection theorem?

What about it?

Then it may well be empty. Are you familiar with the proof of this theorem?