Elementary Topology Problem: What is $\tau_c$?

Question

This is the problem set I given by Munkres Topology (2014, 2nd int'l edition)

Show that the collection $\tau_c$ given in Example 4 of §12 is a topology on the set X where Example 4 of §12 is following picture

Then, is the collection$$\tau _∞=\{U|X−U \text{is infinite or empty or all of }X\} $$ a topology on $X$?

My Question

I can't understand what $\tau_c$ is on Figure 13.4 . It looks $\tau_\infty$ refers to the infinite collection of $\tau_c$, but I am not sure of it either.

Any guidance to understand it right?

Answer 1

Look at the right section! It's the example 4 on page 77, Let $\mathscr{T}_c$ be the collection of all subsets $U$ of $X$ such that $X- U$ is either countable or is all of $X$.