[jdpage]

From a real analysis standpoint... the definition of a 'closed set' in an N-dimensional metric space (of which Euclidean space, i.e. normal space, is an example) is as follows: a set C is 'closed' if and only if, given any sequence of elements (x_n) converging on x, such that (x_n) is a subset of C, it follows that x is also in C.

Under this definition, 'closed' makes sense in the larger context, since in mathematics, if an operation is 'closed' on a set, that means that applying the operation to elements of the set always yields another element of the same set.

Open is a bit more awkward. A set O is 'open' if and only if the complement of O (i.e. the set of all points not in O) is closed. So open is kind of the opposite of closed, hence the convention. Of course, it isn't really the opposite of closed, since there are sets which are neither closed nor open.

tt;dr (too technical; didn't read) in maths, 'closed' usually means 'I can do stuff inside this set without falling out of it'. In the case of intervals, the 'stuff' in question is taking a limit of a convergent sequence.

[GFK_of_xmaspast]

You can also have sets that are simultaneously closed and open.

[monochromatic]

Yep. For example, the empty set or the entire real number line.

[ntlve]

Out of curiosity, do you have any examples or references to something with examples of sets that are neither open nor closed?

[lmkg]

The set of rational numbers that lie inside the interval [0,1].

This set is not closed there are non-rational numbers in that interval which are limit points of sequences that consist only of rationals. For example, any of the algebraic numbers. I think that all real numbers are limits of such sequences, but I might be mis-remembering some subtlety of Dedekind Cuts (one method for constructing the Reals).

This set is not open because any rational is the limit of a sequence of non-rational reals. This probably makes intuitive sense, but just for the sake of formality: To construct such a sequence for any rational r, start with the number x_1 = 1/pi, and approach by a factor of 1/pi at each step, i.e. x_n+1 = x_n + (r-x_n)/pi . x_n is irrational because pi is transcendental.

Any simple interval in R will be either closed or open on each end (but it could be closed on one and open on the other). It's more illustrative to create a set with a non-compact interior. In higher dimensions it's possible to have more exotic borders on an interval, but I think that border will just end up being isomorphic to a non-compact set in a lower dimension.

[jdpage]

The simplest example I can think of is the interval (0, 1].

Proof that it's not closed: the sequence (1, 1/2, 1/4, 1/8, ...) is entirely inside the interval, but converges on 0, which is outside the interval, therefore etc.

Proof that it's not open: the sequence (2, 3/2, 5/4, 9/8, ...) is entirely outside the interval (i.e. inside the complement), but it converges on 1, which is inside the interval (i.e. outside the complement). Thus the complement of the interval is not closed, therefore etc.

[mnembrini]

[0,1) = {x | 0<=x<1} is probably the simplest example.

[gh02t]

Also, it is possible for the opposite to occur. So-called clopen sets are both open and closed.

http://en.m.wikipedia.org/wiki/Clopen_set