Section 52: Problem 1 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
A subspace
of
is said to be star convex if for some point
of
, all the line segments joining
to other points of
lie in
.
(a) Find a star convex set that is not convex.
(b) Show that if
is star convex,
is simply connected.
(a) See Figure 1↓.
(b) Consider
such that
. Given any point
,
as a function of
defines a path from
to
lying entirely in
by the definition of a star convex subspace. Hence,
is path connected. Moreover,
, i.e. the identity map on
, and
, i.e. a constant map, implying that
is contractible. Now, take any path
in
. Then, the composition
defined by
is a continuous function such that
while
, i.e.
is a homotopy between
and the constant map
.