Proof That Projection Mappings From a Plane to a Line Are Continuous With Respect to the Continuous Topology
Define the projection of the point
onto the real
and
axes respectively as
Consider the inverse function
If
acts on an open interval
the result is a vertical strip with
The set of open intervals is a basis for
and the inverse image of each open interval is an open strip as above. These open strips are open in
so
is continuous, and similarly for
