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?
 A: There is a MathOverflow question about this very issue; this answer is a nice intuitive explanation, though you will probably also find some of the other answers useful.
A: From Wikipedia:
In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first introduced by Kazimierz Kuratowski, in a slightly different form that applied only to Hausdorff spaces.
A: 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 $[0,\infty]$.
A: Think of the half-open interval $(0,1]$ with the usual open sets (e.g. $(1−ε,1]$ is an open neighborhood of $1$.
Then modify the collection of sets considered "open" so that every open neighborhood of $1$ contains some set of the form $(1−ε,1]∪(0,ε)$, i.e. it covers small parts of BOTH ends of the interval. Can you understand that this modification in which sets are considered open also modifies the way in which the space is connected together?
