Study of geometric properties preserved under continuous deformations.
These two powerful ideas help describe how a space behaves.
A space is compact if it's "small" in a certain sense: every collection of open sets that covers it can be reduced to a finite collection. Think of it as being able to cover the space with a finite number of "patches."
A space is connected if it's all in one piece. If you can draw a path between any two points without leaving the space, it's connected.
Compactness and connectedness help mathematicians understand the structure of spaces, solve equations, and prove theorems about continuity.
A closed interval [0,1] is compact and connected.
Two separate circles are not connected.
Compactness and connectedness describe how spaces fit together and behave.