Download MCS 451-Exercises Q1. Show that every closed subset of a compact

MCS 451-Exercises
Q1. Show that every closed subset of a compact space is compact.
Q2. Let 𝒯 and 𝒯 ∗ be topologies on 𝑋 with 𝒯 ⊆ 𝒯 ∗ .
a) Show that if 𝒯 ∗ is compact then 𝒯 is compact
b) Is the converse true? Namely, if 𝒯 is compact then can we conclude that 𝒯 ∗ is
compact?
c) Prove that if 𝒯 and 𝒯 ∗ are both compact and Hausdorff, then 𝒯 = 𝒯 ∗
Q3. Let 𝑋 be a topological space and 𝐴, 𝐵 ⊆ 𝑋 compact subspaces.
(1) Is 𝐴 ∪ 𝐵 compact?
(2) Is 𝐴 ∩ 𝐵 compact?
(3) The same question under the assumption that 𝑋 is Hausdorff.
Q4. Let 𝑋 be Hausdorff and 𝐴 ⊆ 𝑋 dense and locally compact. Prove that 𝐴 is open.
Deduce from this that any locally compact subspace of a Hausdorff space is relatively
closed, (i.e. open in its closure).
Q5. Show that the rationals ℚ are not locally compact.
Proof: Let 𝑞 be any rational, with 𝐾 a compact set containing 𝑞. Any neighborhood
of 𝑞 will contain an interval (𝑎, 𝑏) ∩ ℚ around 𝑞, and this neigh borhood, when considered as a subset of ℝ must contain some irrational 𝑡. It is straightforward to construct
a sequence of rationals in (𝑎, 𝑏) ∩ ℚ which converge to 𝑡 ∈ ℝ and, as a subset of the
rationals, clearly have no (rational) limit point. It follows that (𝑎, 𝑏) ∩ ℚ cannot be in
𝐾 as (𝑎, 𝑏) contains an infinite sequence with no convergent subsequence. Hence ℚ is
not locally compact at 𝑞 (or any rational).
Q6. Let 𝑋 be an uncountable set, and let 𝐴 be a countable subset of 𝑋. Show that
𝑋 − 𝐴 is uncountable.
Q7. Consider the set 𝑋 = {𝑎, 𝑏, 𝑐}, and the collection of subsets 𝒯 = {∅, 𝑋, {𝑎}, {𝑏, 𝑐}}.
List the compact subsets of 𝑋. Give an example of a function 𝑓 : 𝑋 → ℝ that is continuous, and one example of a function 𝑔 : 𝑋 → ℝ that is not.
Q8. Give an example of a continuous function on (0, 1) which is bounded but attains
neither a maximum value nor a minimum value.
Q9. Show that a closed subspace of a normal space is normal.
Q10. Show that a closed subspace of a normal space is normal.
Q11. Show that every locally compact Hausdorff space is regular.
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