Section 31: Problem 4 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
and
denote a single set under two topologies
and
, respectively; assume that
. If one of the spaces is Hausdorff (or regular, or normal), what does that imply about the other?
If a space is Hausdorff then it is Hausdorff in a finer topology. There is no such relation for the two other properties. Indeed,
is normal and regular, while
is not even regular. At the same time every space in the discrete topology is normal.