# Comparing product topologies

I'm reading Topology by Munkres, and I'm having some trouble with exercise 5 on page 92 (screenshot attached).

## Homework Statement

Let X and X' denote a single set in the topologies T and T', respectively; let Y and Y' denote a single set in the topologies U and U', respectively. Assume these sets are nonempty.
Show that if T' is finer than T and U' is finer than U, then the product topology on X' x Y' is finer than the product topology on X x Y.

## The Attempt at a Solution

I think this is a counterexample...
Let T = T' = U = U' be the standard topology on ℝ. Let X = Y = Y' = (0, 1). Let X' = (2, 3).

X is open in X and Y is open in Y → X x Y is open in X x Y.
X is not a subset of X' →X is not open in X' → X x Y is not open in X' x Y'.
Hence the product topology on X' x Y' is not finer than the product topology on X x Y.

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So you're dropping to the subset topologies on X,X',Y and Y' right? However, the examples you've chosen are not comparable and so your result is meaningless (check Munkres, he even talks about how some topologies are not comparable). Hence the result only has meaning when you can compare the topologies.

Deveno