Homework 1 Topology I, Fall 2013

Homework 1
Topology I, Fall 2013
Problem 1 (An ultrametric space). Let X be the set of all right-infinite binary words, i.e.,
X = {x1 x2 x3 · · · | xi ∈ {0, 1}, for i = 1, 2, . . . }.
(a) Define dist : X × X → R as follows. For x, y ∈ X, if x 6= y, then
dist(x, y) = e−|x∧y| ,
where x ∧ y denotes the longest common prefix of x and y, and |x ∧ y| denotes its length, and if
x = y, then d(x, y) = 0. Show that d is a metric on X.
(b) An ultrametric on a set Y is a function d : Y × Y → R that satisfies the following laws:
1. (positivity) d(x, y) ≥ 0 with equality if and only if x = y,
2. (symmetry) d(x, y) = d(y, x), and
3. (ultrametric inequality) d(x, z) ≤ max{d(x, y), d(y, z)}.
Note that every ultrametric is a metric (do not show this).
Show that if two balls in a metric space Y with an ultrametric d intersect nontrivially then one
of them is contained in the other.
(c) Show that dist defined in (a) is an ultrametric on X.
Problem 2 (Topology forgets distances). Let (X, d) be a metric space. Define d : X × X → R
and d0 : X × X → R by
d(x, y) = min{1, d(x, y)}
d0 (x, y) =
d(x, y)
1 + d(x, y)
(a) Show that both d and d0 are metrics on X (note that both of these metrics are bounded;
the metric d is called the standard bounded metric induced by d).
(b) Show that the topologies on X induced by d, d and d0 are the same (thus, topology “does
not remember” if a metric space was bounded or not; also, balls of large radius are not important).
Problem 3 (Continuity and closure). Let X and Y be topological spaces and f : X → Y a
function. Prove the following statements.
(a) f is continuous if and only if, for all A ⊆ X, we have f (A) ⊆ f (A).
(b) f is closed if and only if, for all A ⊆ X, we have f (A) ⊇ f (A).
Problem 4 (Limit points have large company (well, sometimes)). Let X be a topological space,
A ⊆ X, and x a limit point of A (every neighborhood of x contains a point from A − {x}).
(a) Show that if X is T1 , then every neighborhood of x contains infinitely many points from A.
(b) Provide an example of a T0 space X and a neighborhood of x that contains only finitely
many points from A.
Problem 5 (Equality is a closed condition (well, sometimes)). Let X and Y be topological spaces
and f, g : X → Y two maps between them.
(a) Show that, if Y is a Hausdorff space, then the set Xf =g = {x ∈ X | f (x) = g(x)} is closed
in X.
(b) Provide an example in which Y is not a Hausdorff space and Xf =g is not closed.
Problem 6 (Gluing Lemma). Let X and Y be topological spaces, X = ∪j∈J Aj , and {fj : Aj →
Y | j ∈ J} a family of maps such that, for all i, j ∈ J for which Ai ∩ Aj 6= ∅, the restrictions
fi |Ai ∩Aj and fj |Ai ∩Aj coincide. Define a function f : X → Y by gluing, i.e., whenever x ∈ Aj let
f (x) = fj (x).
Show that f is continuous under either of the following two conditions.
(a) Each Aj is open.
(b) Each Aj is closed and J is finite.
Problem 7 (If all you ask is T1 , that’s what you get). Let X = (R, τ ), where τ is the smallest
topology in which every singleton {x}, for x ∈ R, is closed. Show that X is a T1 space, but not a
Hausdorff space.