Disclaimer: There are many topologists here and they may not like the philosophical flavour of my answer :-)
I think it all starts with the end... actually with the notion of "end". Take an open interval (a,b). It is bounded, yet you cannot reach its ends! At first this may look weird, but then one realizes this weirdness is to be attributed to the mathematically exact observation of this object (mathematicians can distinguish between so many things like point, set of points, boundedness, boundary etc.). Encountering such "weirdness" only shows the strong need for an exact and abstract formulation of "having no end". The lack of ends we then call "openness". If we are looking for the best generalization of this concept, the first approach would be of course a set-theoretical one. And in fact, in the case of intervals it turns out that the property "having no ends" is inhereted into arbitrary unions and finite intersections. Any attemps to expand this lead either to contradictions to our basic example or to unjustified reduction in generality.