a topological space
Imagine you want to talk about 'nearness' without ever measuring a distance. On a map you can say two towns are close; on a rubber sheet you can stretch the sheet so distances change wildly, yet which points sit near which others — in the sense of being surrounded together — can stay the same. A topological space is the bare minimum of structure you need to make that idea of 'being near' precise, with no ruler in sight. It is geometry boiled down to the question: which points cling together no matter how you bend the picture?
Here is the actual machinery. You start with a set X of points, and you single out a family of subsets of X, called the open sets, that you declare to be 'open'. This family must obey just three rules: the whole set X and the empty set are open; any union of open sets (however many) is open; and the intersection of finitely many open sets is open. That is the entire definition — the chosen family is called a topology on X. Everything topology studies (continuity, connectedness, holes) is squeezed out of those three rules. For example, on the ordinary line, the open sets are exactly the sets you can write as a union of open intervals (a, b); 'near a point x' then means 'inside some open interval around x'.
Why so spare? Because by throwing away distance and keeping only the open sets, you keep exactly the features that survive stretching and bending — and those are the features two shapes share when one can be continuously deformed into the other. A circle and a square are different metrically (one has corners, one does not), but as topological spaces they are the same. The price is that topology cannot see size, angle, or curvature at all; it is deliberately blind to them, which is its power and its honest limitation.
Take X = {a, b, c} with the three points and declare the open sets to be exactly { }, {a}, {a, b}, {a, b, c}. Check the rules: the empty set and the whole set are listed; any union of these is again on the list; and any intersection of finitely many is too. So this is a valid (rather lopsided) topology on three points — proof that a topological space need not look anything like a smooth shape.
A finite, non-geometric topology: the open sets, not the picture, are what define the space.
A 'topology' on a set is the chosen family of open sets, not the subject 'topology' — and the same set X can carry many different topologies, each making it a different topological space.