Summary of Equiv. Statements Regarding Cts. Maps on Topo. Spaces

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:

• Continuous Maps on Topological Spaces

• The Open Neighbourhood Definition of Continuous Maps on Topological Spaces

• Equivalent Statements Regarding Continuous Maps on Topological Spaces

• The Closed Set Definition of Continuous Maps on Topological Spaces

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.