TITLE: Why is this definition of topological space equivalent to the common one?
QUESTION [6 upvotes]: While comparing the definition of a topological space among books, I found one book [1]  whose definition seems to differ from the others. Here it is:
Definition: Let $X$ be a non-empty set. A class (defined by the book as a "set of sets") $T$ of subsets of $X$ is called
a topology on $X$ if it satisfies the following two conditions:

The union of every class of sets in $T$ is a set in $T$
The intersection of every finite class of sets in $T$ is a set in $T$

This definition omits the usually present condition that the empty set and $X$ itself must be in $T$. It also adds the condition that $X$ is not empty. Shortly after this definition, the author writes:

We observe that the empty set and the full space are always open sets
in every topological spaces, since they are the union and intersection
of the empty class of sets, which is a subclass of every topology.

Which doesn't make sense to me, given the definition.

<deleted>
Why is $X$ in $T$?
Why does $X$ need to be non-empty?

Update (Much later):
I discovered that the book adopts a convention by which all sets under discussion are restricted to $X$. Not realizing/mentioning that was entirely my fault.
[1] George F. Simmons - Introduction to Topology and Modern Analysis, pp. 92

REPLY [6 votes]: When dealing with collections of subsets of a common superset $\Omega$, intersections and unions are usually defined as follows:
Let $\mathcal B$ be a collection of subsets of $\Omega$, then
\begin{align}
\bigcup \mathcal B &:= \bigcup_{A\in\mathcal B} A := \left\{\, x\in\Omega \,\middle|\, \exists A\in\mathcal B\colon x\in A \,\right\} \subseteq \Omega, \\
\bigcap \mathcal B &:= \bigcap_{A\in\mathcal B} A := \left\{\, x\in\Omega \,\middle|\, \forall A\in\mathcal B\colon x\in A \,\right\} \subseteq \Omega.
\end{align}
For $\mathcal B=\varnothing$ this yields $\bigcup \mathcal B = \varnothing$ and $\bigcap \mathcal B = \Omega$.
For $\mathcal B\neq\varnothing$ the definitions above don't depend on $\Omega$ and agree with the usual intersections and unions of collections of sets. Even the union $\bigcup \varnothing$ doesn't depend on $\Omega$, the only problem is the empty intersection $\bigcap \varnothing=\Omega$, which is a very useful definition in the context of a common superset.
You can think of $\varnothing$ as the identity element with respect to taking unions and of $\Omega$ as the identity element with respect to taking intersections of subsets of $\Omega$. This makes the empty union and empty intersection analogous to the empty sum $\sum_{a\in\varnothing} a = 0$ and the empty product $\prod_{a\in\varnothing} a = 1$, which are the identity elements of addition and multiplication, respectively.
In the case of a topology, $\Omega=X$ and $\tau$ is a collection of subsets of $X$. Now $\varnothing\subseteq \tau$ is vacuously true, since every element of $\varnothing$ is an element of $\tau$: there is no element in $\varnothing$ that could serve as a counter example. Hence, the two axioms imply $\bigcup\varnothing = \varnothing\in\tau$ and $\bigcap \varnothing= X\in\tau$.
Requiring $X$ to be non-empty is indeed unusual, maybe the author was too lazy to exclude the empty space from some of the theorems.