Topology, Pasting Continuous Functions Together
Pasting Continuous Functions Together
Let a collection of open sets cover the domain of a function f,
and assume f is continuous on each open set.
Let R be open in the range and let Q be the preimage.
Let W be an open set in the cover and let V = W∩Q.
Since f is continuous on W, V is open in W.
And W is open in the domain, hence V is open in the domain.
There is a V for each W in the cover, and their union gives Q.
Hence the preimage of R is open and f is continuous.
If the domain is covered by closed sets, and the cover is locally finite,
we can make the same claim.
The preimage of a closed set is divided into chunks,
a closed piece V for each W in the cover.
The collection V[i] is locally finite,
and its union is closed.
The preimage of a closed set is closed, and f is continuous.