means The branch of mathematics studying the properties of shapes that survive stretching, bending, and twisting—but not tearing or gluing—so it cares about holes and connectedness rather than exact measurements.

from From the Greek 'topos' (place) plus '-logia' (study of), so literally 'the study of place.' The term entered mathematics in the 19th century; an earlier name was the Latin 'analysis situs,' meaning 'analysis of position.' The same 'topos' gives us 'topic' and 'utopia' (literally 'no-place').