a homeomorphism
/ HOH-mee-oh-MOR-fizm /
This is the rubber-sheet equals sign. Two shapes count as 'the same' in topology if you can deform one into the other by stretching, shrinking, and bending, but never tearing and never gluing. The famous slogan is that a coffee cup equals a doughnut: the cup's handle is the doughnut's hole, and you can mould one into the other through a smooth, reversible squashing with no cuts. A homeomorphism is the precise object that certifies two spaces are equal in exactly this sense.
Precisely, a homeomorphism between spaces X and Y is a function f from X to Y that is (i) a one-to-one correspondence — every point of Y is hit exactly once, so f has an inverse — and (ii) continuous both ways: f is continuous and its inverse is continuous too. Continuity here is the topological kind: f is continuous when the preimage of every open set is open, which captures 'nearby points go to nearby points' without any ruler. The two-way condition matters: it forbids any cut (which would tear neighbours apart) and any glue (which would crush distinct points together). When such an f exists we say X and Y are homeomorphic, written X is congruent-to Y in spirit but topologically equal.
Homeomorphism is the equivalence relation that defines the whole subject: topology is precisely the study of properties unchanged by homeomorphisms — the topological invariants, like number of holes, connectedness, and compactness. The crucial honesty is that homeomorphic is far weaker than congruent or even than the metric notion isometric. A coffee cup and a doughnut are homeomorphic but plainly not the same size, shape, or curvature; a tiny circle and a vast circle are homeomorphic. So 'topologically the same' tells you about holes and connectivity, and nothing whatsoever about distance, angle, or size.
Map the open interval (-1, 1) to the whole real line by f(x) = x / (1 - x^2), or just as cleanly by f(x) = tan(pi x / 2). Both are continuous, one-to-one, with continuous inverses, so the bounded interval (-1, 1) is homeomorphic to the entire infinite line — even though one is 'short' and the other is endless. Length is not a topological property.
A finite open interval is topologically identical to the infinite line; size is invisible to topology.
Both directions must be continuous: a continuous one-to-one map need not be a homeomorphism (wrapping the line onto a circle is continuous and onto small pieces, but unrolling can tear), and homeomorphic never implies same size or shape.