Download Topology I – Problem Set Two Fall 2011

Topology I – Problem Set Two Fall 2011
1. Definition: A directed set is a non-empty partially ordered set D with the property that
for any a, b ∈ D there is a c ∈ D such that c ≥ a and c ≥ b. Let X be a topological
space. A net in X is a function f : D → X from a directed set to X.
E.g. If D = N, the natural numbers with the usual ordering, then a net in this case is
just a sequence in X.
Definition: A net f : D → X is said to converge to x ∈ X if for every neighborhood U
of x, there is a d ∈ D such that a ≥ d implies f (a) ∈ U .
Prove:
(a) A subset A ⊂ X is closed if and only if there is no net in A converging to a point
not in A.
(b) A space X is Hausdorff if and only if each net converges to at most one point.
(c) A function f : X → Y is continuous if and only if for every net g converging to a
point x ∈ X, the net f ◦ g converges to f (x).
2. Let X be a T1 topological space. Prove that X is countably compact (as defined in
Schaum’s outlines) if and only if every countable open cover has a finite subcover.
P
3. Let l2 denote the set of square summable sequences, i.e. l2 = {(a1 , a2 , . . . )| a2i < ∞}.
The following is a metric on l2 (you don’t have to prove this):
qX
(ai − bi )2 .
d((a1 , a2 , . . . ), (b1 , b2 , . . . )) =
Show that the unit ball B = {(a1 , a2 , . . . ) ∈ l2 |
that B is not sequentially compact).
P
a2i ≤ 1} is not compact (Hint: show
4. A topological space is called separable if it contains a countable dense subset. A topological space is called Lindelof if every open cover contains a countable subcover. A
topological space is called second countable if it has a countable base for the topology.
Prove:
(a) A space that is second countable is separable.
(b) A space that is second countable is Lindelof.
(c) A separable metric space is second countable.
5. Let Rl be the real line with the following topology: A base consists of the set of half-open
intervals [a, b). The space Rl is called the lower limit topology, or the half-open interval
topology, or sometimes the Sorgenfrey line.
(a)
(b)
(c)
(d)
Show
Show
Show
Show
that
that
that
that
Rl is not connected (Hint: each base element is open and closed).
Rl is seperable but not second countable, hence not metric.
Rl is Lindelof.
Rl × Rl is separable but not Lindelof.