topological space
Think of a topological space as the bare skeleton of geometry: you forget all notion of distance, angle, and size, and you keep only one piece of information — which sets count as “open”, that is, which sets are “roomy” enough that every point inside has a little wiggle room around it. From this single ingredient you can still talk about nearness, continuity, and connectedness, even though there are no numbers measuring how far apart points are.
Formally, a topological space is a pair (X, T) where X is a set and T is a collection of subsets of X, called the open sets, satisfying three axioms: the empty set and X itself are open; any union (even infinitely many) of open sets is open; and any finite intersection of open sets is open. The collection T is called the topology on X.
A metric space gives a topological space for free — call a set open if every point in it sits inside some open ball contained in the set — but the converse fails: there are topological spaces whose topology comes from no metric at all. Topology is therefore strictly more general than metric geometry, which is exactly why it isolates the properties that survive bending and stretching without tearing.
On X = {a, b, c} the family T = { {}, {a}, {a, b}, {a, b, c} } is a topology: it contains {} and X, and is closed under the unions and finite intersections you can form from it. Here {a} is open but {c} is not, so a and c sit in the space very differently.
A topology on a three-point set need not be discrete; finiteness makes the axioms easy to check by hand.
The same set X can carry many different topologies. At one extreme is the discrete topology (every subset is open), at the other the indiscrete or trivial topology (only the empty set and X are open). Comparing topologies on a fixed set — which is finer, which is coarser — is a recurring theme.