Topology Seperation Axioms Let f g X rightarrow Y be contin

Topology - Seperation Axioms

Let f, g: X rightarrow Y be continuous; assume that Y is Hausdorff. Show that {x | f (x) = g(x)} is closed in X.

Solution

Given that Y is Hausdorff. That means two distinct points in Y has two distinct neighbourhoods.

Let us consider x an element in Y.

f(x) = g(x)

Since f, g are continuous and Y is Hausdorff,

f, g have both its limit points in x

limt x tends to a f(x) = c is not in Y

then x has a neighbourhood in Y not having c

But this contradicts the fact that Y is Hausdorff.

Hence X contains all its limit points or x such that f(x) = g(x) is closed