Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2015
Math 636 — Problem Set 2
Issued: 09.11
Due: 09.18
2.1. Show that a subspace of a Hausdorff space is Hausdorff.
2.2. Let X be the plane R2 with the basis of topology given by the usual
open sets together with the sets {(x, y) : x2 + y 2 < a, y 6= 0} ∪ {(0, 0)}.
Show that this space is Hausdorff but not regular.
2.3. Suppose that A is a connected subset of a topological space, and
A ⊂ B ⊂ A. Show that B is connected.
2.4. Suppose that a metric space (X, d) is connected. Show that for every
x, y ∈ X and > 0 there exists a sequence x0 = x, x1 , x2 , . . . , xn = y
such that d(xi , xi+1 ) < for all i = 0, 1, . . . , n − 1. Is the converse true?
2.5. Erdős’ example. P
Let `2 be the set of all sequences (x1 , x2 , . . .) of real
2
numbers such that ∞
n=1 xn < ∞. It is known that d((x1 , x2 , . . .), (y1 , y2 , . . .)) =
P∞
1/2
( n=1 (xi − yi )2 ) is a metric on `2 . We denote by kwk, for w ∈ `2 ,
the distance from w to the constant zero sequence.
Let X be the subset of `2 consisting of sequences of rational numbers.
Show that X is totally disconnected.
Consider the set V = {w ∈ X : kwk < 1}, and let U ⊂ V be a neighborhood of (0, 0, . . .). Show that U \U is non-empty, and hence X is not
zero-dimensional. (Hint: find a sequence wk = (a1 , a2 , . . . , ak , 0, 0, . . .) ∈
U such that distance from wk to X \ U is not more than 1/k.)
2.6. Local connectivity. A topological space X is called locally connected
if for every point x ∈ X and every open neighborhood U of x there
exists a connected neighborhood V of x such that V ⊂ U . Give an
example of a connected but not locally connected subset of R2 .