Download TOPOLOGY PROBLEMS MARCH 20, 2017—WEEK 5 1. Show that if

TOPOLOGY PROBLEMS
MARCH 20, 2017—WEEK 5
1. Show that if Y is compact, the projection map p1 : X × Y → X is closed; that is, the image
of any closed set is closed. (Hint: Tube Lemma.)
2. We say a space X is countably compact if every countable (spočetná) open cover of X contains
a finite subcover. Notice that every compact space is countably compact.
(i) Show that a space X is countably compact iff every nested sequence of nonempty closed subsets
C1 ⊇ C2 ⊇ · · · ⊇ Cn ⊇ . . . of X has nonempty intersection.
(ii) Challenge: find a topological space that is countably compact, but not compact.
3. We say a space X is limit point compact if every subset S ⊆ X has a limit point: some x ∈ X
such that every neighborhood of X contains a point of S (not equal to x itself).
(i) Show that every compact space is limit point compact.
(ii) Consider the space X = Z+ × Y , where Z+ has the subspace topology from R and Y is a two
point space with the indiscrete topology. Show that X is limit point compact, but not compact.
(iii) Show that a T 1 space is countably compact iff it is limit point compact.
4. Let X be limit point compact.
(i) If Y is limit point compact as well, is X × Y ?
(ii) If f : X → Y is continuous, is f (X) limit point compact?
(iii) If A is a closed subspace of X, is it limit point compact?
(iv) If X is a subspace of a Hausdorff space Y , is it closed?
5. Let X be a linearly ordered set with the order topology: the base of open sets consists of all
intervals (a, b), [a0 , b), and (a, b0 ), where a0 and b0 are, respectively, the minimum and maximum
elements of X (if they exist).
(i) Show that if every closed interval in X is compact, X has the least upper bound property: if
S ⊆ X has an upper bound in X, then it has a least upper bound (supremum) in X.
(ii) Show that if X is a linear order with the least upper bound property, every closed subset is
compact. (Hint: Munkres, 27.1.)
6. (Extreme Value Theorem) Let X be a topological space, Y a linearly ordered set in the order
topology, and f : X → R a continuous function. Prove that f attains its maximum and minimum
values on any nonempty compact subspace of X.
7. Let f : X → Y be a closed continuous surjection that is perfect: for any y ∈ Y , f −1 ({y}) is
compact in X. Show that if Y is compact, so is X.
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