Section 25*: Problem 5 Solution
Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
Let
denote the rational points of the interval
of
. Let
denote the union of all line segments joining the point
to points of
.
(a) Show that
is path connected. Show that every connected open set in
is path connected.
(b) Find a subset of
that is path connected but is locally connected at none of its points.
(a) Every point is (path) connected to
, hence, the space is (path) connected. For the same reason it is locally (path) connected at
.
But for any other point every small enough neighborhood consists of disjoint line segments and is not (path) connected. More formal work is needed here. As I commented below it is not enough that the subspace is a union of disjoint line segments to claim that it is not connected. You need to show that there is a neighborhood of the point such that it contains no connected subneighborhoods. For this, a) choose a small ball at the given point that does not contain
, consider any its subneighborhood, and choose a ball at the point contained in this neighborhood, b) choose an irrational point
in
such that
splits the (smaller) ball into two non-empty parts, and c) basically you are done: this line segment then splits the given neighborhood into two non-empty parts.
(b) The idea of the example in (a) is that every path connecting two points lying on two different line segments goes through a given point, and, moreover, at any other point every neighborhood intersects at least two (and hence an infinite number of) different line segments, so that a small enough neighborhood that does not contain the given point is not connected. To make the space not connected at
as well, we can make a copy of the space in such a way that for the copy another point
plays the role of
while
is also in that copy of the space but plays the role of “another point” .
Let
, then
is (path) connected (as the union of the path connected line segment
and the collection of other path connected line segments, each of which intersects
) but not locally connected at any point (now it is not locally connected at any point including
or
using the argument as above).