This is my check that $\mathcal{T}_c$ is a topology in James R. Munkers’: Topology 2nd Edition, Section 12, Example 4.

Definition of a Topology

A topology on an arbitrary set $X$, denoted by $\mathcal{T}_c$ in this case, is a collection of the subsets of $X$ adhering to three properties:

1) $\emptyset$ and $X$ are in $\mathcal{T}_c$
2) The union of the elements of any nonempty subcollection of $\mathcal{T}_c$ is in $\mathcal{T}_c$
3) The intersection of the elements of any nonempty, finite subcollection of $\mathcal{T}_c$ is in $\mathcal{T}_c$

$\mathcal{T}_c$ needs to adhere to these three rules.

$\mathcal{T}_c$ definition

Let $\mathcal{T}_c$ be the collection of all subsets $U$ of $X$ such that $X - U$ either is countable or is all of $X$.

Checking the properties for a topology

1) $X - \emptyset = X$ as $\emptyset$ is a subset of $X$. $X - X = \emptyset$ as all of $X$ is a subset of $X$.
2) If ${U_\alpha}$ is an indexed family of nonempty elements of \(\mathcal{T}_c\), need to show $\bigcup U_\alpha$ is in $\mathcal{T}_c$, compute

\[X - \bigcup U_\alpha = \bigcap(X - U_\alpha)\]

By De Morgan’s Law

3) TBD until I understand how the intersection of all $X - U_\alpha$ proves property number 2 for a topology.

Extras

Link to the mathjax LaTeX specification: https://treeofmath.github.io/tex-commands-in-mathjax/TeXSyntax.htm