If (X, T ) is a locally compact Hausdorff space, show that X , as a subset of its Tychonoff-C? ech compactification K , is open (if f is the homeo- morphism of X into K given in the definition of Tychonoff-C? ech com- pactification, the range of f is open). Hint: Given any x ∈ X , let U be a neighborhood of x with compact closure. Show that there is a continuous real function f on X ,0 at x , and 1 on X \U . Use this function to show that x is not in the closure of K \U.