Why not choose a definition of a topology that rules out $\tau = \{\varnothing\}$?

Question

When defining a topology via the open set definition, why do we not insist that $\varnothing \neq X$?

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The open set definition of a topology includes the following things:

1. $\varnothing$ is an element of $\tau$.
2. $X$ is an element of $\tau$.
3. If $\mathcal{F}$ is a family of elements of $\tau$, then $\cup \mathcal{F}$ is an element of $\tau$ too.
4. If $A$ and $B$ are elements of $\tau$, then $A \cap B$ is an element of $\tau$.

Topologies cannot be turned into a first-order theory, but conditions (1) and (2) are reminiscent of the existence of $0$ and $1$ in fields, and in that setting we insist that $0 \ne 1$ which rules out interesting psuedofields like the field with one element.

The above analogy is extremely loose, but I'm curious why we don't do the analogous thing for topologies and insist that $\varnothing \neq X$.

Answer 1

Maybe because the empty set (considered as a topological space) is the initial object of the category of topological spaces.

Answer 2

I guess that you might be correct, that if empty set is allowed as a topological space, then it makes sense to also allow the zero ring as a legitimate commutative unital ring. But, a field should still be defined as having $0 \neq 1$. So the zero ring should not be considered a field. This is analogous to how $1$ is not a prime number. For example, you want unique decomposition into primes, and if you set $1$ to be a prime number you don't have it (but if not, then very elegantly $1$ decomposes to the empty product of primes, so everything is well). So fields correspond to "possible atoms out of which a spectrum is built", and they should be non-trivial.