Topology M.S. exam. U = int( ¯ U ).

Topology M.S. exam.
1. Prove or disprove: If U is an open set then
U = int(Ū ).
Here int(U ) denotes the interior and Ū denotes the closure of the set U .
2. Show that every compact Hausdorff space is normal.
3. A relation ∼ on X is an equivalence relation if
1. x ∼ x for all x ∈ X;
2. if x ∼ y then y ∼ x for all x, y ∈ X;
3. if x ∼ y and y ∼ z then x ∼ z for all x, y, z ∈ X.
Let X be a connected space. Suppose that ∼ is an equivalence relation on
X with the property that, for each x ∈ X there is an open neighborhood Ux of
x such that
for all
y ∈ Ux .
Prove that x ∼ y for all x, y ∈ X.