One of the comments on the original post said that you can define a topology in terms of neighbourhoods. I'd like to amplify on that comment because it's the answer I favour too, if you want to do things as intuitively as possible. In fact, you can do it with basic open neighbourhoods, which is often nicer, for reasons I'll come to in a moment.
The first step would be to axiomatize the notion of a basic open neighbourhood. So it would consist of properties like that if N is a b.o.n. of x then x is an element of N, that if y is also an element of N then there is a b.o.n. N' of y such that N' is a subset of N (and in many systems one would be able to take N'=N), that the intersection of two b.o.n.s of x is another one, and so on. Suppose we've got all that sorted out. Then the rest of the definitions are just like metric space definitions without the need to reformulate those definitions in terms of open sets. To give the most important example, a function $f:X\to Y$ is continuous at x if and only if the following condition holds: for every b.o.n. M of f(x) there exists a b.o.n. N of x such that $f(N)\subset M$. Of course, in a metric space the basic open neighbourhoods of x are the open balls $B_\epsilon(x)$.
In the usual definition of continuity for maps between topological spaces, one never talks about continuity at a point, but it is perfectly possible and natural to do so, as the above shows.
Here's another example: a set F is closed if and only if for every x not in F you can find a basic open neighbourhood N of x that is disjoint from F. Oh, and I should have said that a set U is open if and only if for every x in U you can find a basic open neighbourhood N of x such that $N\subset U$.
The one thing you can't do is reformulate these definitions in terms of sequences, for the simple reason that the sequence reformulations do not generalize to topological spaces (unless you replace them by nets).
Added later: I've just seen some more of the comments on the original post. Much of what I have said is implicit in those comments, but perhaps it is useful to have it spelt out.