Section 16: Problem 2 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
If
and
are topologies on
and
is strictly finer than
, what can you say about the corresponding subspace topologies on the subset
of
?
Let
and
be the subspaces of
and
, respectively. Then,
is finer but not necessarily strictly finer than
. It is finer, because if we change the topology on
from
to
then all subsets of
that were open are still open, and therefore their intersections with
are still open in
. It is not necessarily strictly finer as the new open sets from
may not produce new open sets in the subspace topology. For example, a one-point subset
of any topological space
always have the same subspace topology regardless of the topology on
.