"Topology" as a word that refers to the shape of something's surface. Topological maps use whorls and markings to indicate elevation and mark hills, ridges, valleys and other features. They're also called "topographical."
It has a math-related meaning as well, which involves the study of those surfaces and ways to express them in mathematical terms. Which is where it gets weird, because the same equations can describe two apparently different shapes. It's a wee bit of a head-scratcher, because our eyes tell us we're looking at two different things, but the math says they're interchangeable. This picture at the Curiosa Mathematica blog models how that can be by showing how a teacup and a donut are topologically equivalent:
Of course, the teacup does not taste nearly as good when glazed, and it's awful hard to sip your Earl Grey from a donut. Facts which mathematicians freely acknowledge.