# Summary of Equivalent Statements Regarding Continuous Maps on Topological Spaces

We have see many different definitions/equivalent statements for a map $f : X \to Y$ (where $X$ and $Y$ are topological spaces) to be continuous on all of $X$. Be sure to review the following pages:

We will now summarize all of the equivalent definitions for $f : X \to Y$ to be continuous on all of $X$ with the following diagram.