Download Topology III Exercise set 1 1. Let Z be a regular space and let z 1,z2

Topology III
Exercise set 1
1. Let Z be a regular space and let z1 , z2 , . . . be distinct points of Z. Assume that
the subspace {z1 , z2 , ...} of Z is discrete. Show that there exist Vn ∈ ηzn (Z), for
n = 1, 2, . . . , such that Vn ∩ Vk = ∅ whenever n 6= k.
2. In the beginning of Example I.2.2, the space Z is described in terms of nbhd bases
of its points. Verify that the description of nbhd bases is correct and that it does,
indeed, define a topological space.
3. Let X be a T1 -space. The double D(X) of X is defined as follows. The ground set
of D(X) is X × {0, 1}, and the topology of D(X) is defined by a base formed by all
sets of the form (G × {0, 1}) r {(x, 1)}, where x ∈ G ⊂◦ X and all singletons {(x, 1)},
where x ∈ X.
(a) Verify that the given sets do, indeed, form a base.
(b) Show that D(X) is T1 , and that D(X) is T2 provided that X is T2 .
(c) Show that if X is a Tihonov space, then so is D(X).
(d) Show that if X is compact, then so is D(X).
4. A space X is extremally disconnected if G ⊂◦ X for every G ⊂◦ X. Show that X is
extremally disconnected iff any two disjoint open subsets of X have disjoint closures.
5. Show that an extremally disconnected T3 -space has no non-trivial convergent sequences.
6. Show that a mapping f : X → Y is closed iff for every y ∈ Y and for each G ∈
ηf −1 {y} (X), there exists O ∈ ηy (Y ) such that f −1 (O) ⊂ G.