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My answer will not be of philosophical nature, neither historical, but perhaps pedagogical.

I find Munkress' Topology a great book. Among other merits, because its introduction, which I summarize as follows:

1. You recall what a metric space is. Define open balls and subsequently, open sets. Prove that, in a metric space:

1.1 The empty set and the total space are open sets.

1.2 The union of an arbitrary number of open sets is an open set.

1.3 The intersection of a finite number of open sets is an open set.

2. Recall what a continuous map between metric spaces is (the $\epsilon$-$\delta$ definition). Prove the theorem that says that a map between metric spaces $f: X \longrightarrow Y$ is continuous if and only if $f^{-1} (U) \subset X$ is an open set for every open set $U \subset Y$.

And you have a motivation for the definition of topological space and continuous map as well.

Of course this is not an historical explanation of how topological spaces arised, nor does it justify why you chose these properties of open sets in metric spaces and not others: "experience" has told us that these are the good ones. (For instance, if I'm not wrong, when Hausdorff first defined topological spaces included the property of being... Hausdorff among the axioms. "Experience" -and not an a priori argument- showed us that it could be interesting to work with non-Hausdorff topological spaces.)