Compact spaces: Tutorial problems
1. Show that A = {(x, y) ∈ R2 : x2 + sin y ≤ 1} is not compact.
2. Show that B = {(x, y) ∈ R2 : x4 − 2x2 + y 2 ≤ 3} is compact.
3. Show that the union of two compact spaces is compact.
4. Find a topological space (X, T ) and a compact subset A ⊂ X such
that A is not closed in X. Can such a space X be Hausdorff?
5. Suppose that X is compact and let f : X → (0, ∞) be continuous.
Show that 1/f is bounded. Does this hold when X is not compact?
6. Let Cn be a sequence of nonempty, closed subsets of a compact
space X suchTthat Cn ⊃ Cn+1 for each n and let A be an open set
that contains Cn . Show that A contains Ck for some k.
Compact spaces: Some hints
1. The set A is unbounded because (0, y) ∈ A for all y ∈ R.
2. Completing the square gives (x2 − 1)2 + y 2 ≤ 4. Use this fact to
show that B is bounded and then show that B is also closed.
3. Suppose A, B are compact subsets of X. If some open sets cover
their union, then these open sets cover both A and B.
4. The space X cannot be Hausdorff because compact in Hausdorff is
closed. As a simple example, let X = {1, 2} and let T = {∅, X}.
5. When X is compact, 1/f attains a min and a max, so it is bounded.
When X is not compact, 1/f is not necessarily bounded.
6. Show that A and the sets X − Cn form an open cover of X. This
implies that X = A ∪ (X − Ck ) for some k, hence also Ck ⊂ A.