Proof That Projection Mappings From a Plane to a Line Are Continuous With Respect to the Continuous Topology

Define the projection of the pointonto the realandaxes respectively as

Consider the inverse functionIfacts on an open intervalthe result is a vertical strip with

The set of open intervals is a basis forand the inverse image of each open interval is an open strip as above. These open strips are open insois continuous, and similarly for

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