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Here's a boring reason, and it may or may not convince you: any function $f : X \to Y$ between topological spaces has the property that the preimage of the entire space $Y$ is the entire space $X$, and the preimage of the empty subset of $Y$ is the empty subset of $X$. So if you allow topological spaces in which either the entire set or the empty set is not open, there are no continuous functions from these spaces to "classical" topological spaces! Given that you agree with me that this is undesirable behavior, I think you are forced to make the entire set and/or the empty set either always open or always not open, and I think if you pick the second option then nothing changes except that, as KConrad says, it becomes unnecessarily harder to say things.

Actually, the situation is even worse: if the empty set isn't allowed to be open in $X$ then the continuous functions $X \to Y$ must be surjective, cannot miss any open set in $Y$, and if the entire set isn't allowed to be open in $X$ then (for a reasonable choice of $Y$) the continuous functions $X \to Y$ must be nonconstant. cannot take values entirely in a proper open subset of $Y$. I think these are both much more unnatural than allowing the entire set and the empty set to be open. This is assuming you agree that the standard definition of continuity is natural.

Here's a boring reason, and it may or may not convince you: any function $f : X \to Y$ between topological spaces has the property that the preimage of the entire space $Y$ is the entire space $X$, and the preimage of the empty subset of $Y$ is the empty subset of $X$. So if you allow topological spaces in which either the entire set or the empty set is not open, there are no continuous functions from these spaces to "classical" topological spaces! Given that you agree with me that this is undesirable behavior, I think you are forced to make the entire set and/or the empty set either always open or always not open, and I think if you pick the second option then nothing changes except that, as KConrad says, it becomes unnecessarily harder to say things.
Actually, the situation is even worse: if the empty set isn't allowed to be open in $X$ then the continuous functions on $X$ X \to Y$must be surjective, and if the entire set isn't allowed to be open in$X$then (for a reasonable choice of$Y$) the continuous functions on$X$X \to Y$ must be nonconstant(unless $Y$ has no open sets whatsoever!). I think these are both much more unnatural than allowing the entire set and the empty set to be open. This is assuming you agree that the standard definition of continuity is natural.
Here's a boring reason, and it may or may not convince you: any function $f : X \to Y$ between topological spaces has the property that the preimage of the entire space $Y$ is the entire space $X$, and the preimage of the empty subset of $Y$ is the empty subset of $X$. So if you allow topological spaces in which either the entire set or the empty set is not open, there are no continuous functions from these spaces to "classical" topological spaces! Given that you agree with me that this is undesirable behavior, I think you are forced to make the entire set and/or the empty set either always open or always not open, and I think if you pick the second option then nothing changes except that, as KConrad says, it becomes unnecessarily harder to say things.
Actually, the situation is even worse: if the empty set isn't allowed to be open then the continuous functions on $X$ must be surjective, and if the entire set isn't allowed to be open then the continuous functions on $X$ must be nonconstant (unless $Y$ has no open sets whatsoever!). I think these are both much more unnatural than allowing the entire set and the empty set to be open.