Alternative definition for topological spaces? I have just started reading topology so I am a total beginner but why are topological spaces defined in terms of open sets? I find it hard and unnatural to think about them intuitively. Perhaps the reason is that I can't see them visually. Take groups, for example, are related directly to physical rotations and numbers, thus allowing me to see them at work. Is there a similar analogy or defintion that could allow me to understand topological spaces more intuitively?

In "Quantales and continuity spaces" Flagg develops the notion of a metric space where the distance function takes values in a value quantale. A value quantale is an abstraction of the properties of the poset $[0,\infty]$ needed for 'doing analysis'. It is then showed that every topological space $X$ is metrizable in the sense that there exists a value quantale $V$ (depending on the topology on $X$) such that the topological space $X$ is given by the open balls determined by a metric structure on $X$ with values in $V$. At this level of abstraction it is thus seen that the open sets axiomatization for topology is nothing but the good old notion of a metric space, only taking values in value quantales other than $[0,\infty]$.
