Topological Spaces and Continuity

Our intuitive understanding of continuity is that, a function is continuous at when points close to are mapped to points close to When we have a notion of distance, for points which get closer and closer to x, the corresponding values of a function should get closer and closer to . Thus, we can formulate the notion of continuity in terms of sequential limits.

There is an equivalent definition due to Weierstrass. It requires the values of the function, for sufficiently close points to , to stay as close to as we want. Although in this case, the definition turns out to be equivalent, the nature of definition as such is different. In the former notion, with sequential limits, the requirement is stated in terms of some set of points( convergent sequences) in the domain to be mapped to suitable set(convergent sequences) in the codomain. Hence the reference is first made to objects in the domain. Whereas in the latter case, we begin with any neighbourhood of in the codomain and require that the inverse image contains a neighbourhood of .

There are several other notions of continuity, not necessarily equivalent, which try to capture some of the intuitive aspects of continuity like a continuum, infinitesimal changes, zero oscillations etc. On the whole, the definition using convergent sequences is more convincing to me.

But to generalize these notions for arbitrary sets, we need to define closeness. So we come up with notions of open sets and neighbourhoods, generalizing the notions of balls, open and closed sets in metric spaces, by requiring them to satisfy some axioms and define a continuous function as a function such that for every open set containing , the inverse image is open. There is an equivalent definition in terms of closed sets. So Topological spaces, which are the context in which we speak of continuous functions, have various characterisations in terms of  open sets, closed sets, neighbourhoods etc. In all these characterisations the definition of continuity is of the same nature as that of Weierstrass’s definition.

It turns out that a function between two topological spaces is continuous iff it maps convergent nets to convergent nets respecting their limits. We had to consider convergent nets for we deal with arbitrary spaces. But we can replace nets with convergent sequences in the above theorem for first-countable spaces. Thus we can think of a topological space as a set with additional structure and continuous functions to be structure-preserving maps.

There is another characterisation of topological spaces in terms of closure operator, given by Kuratowski. In this, every subset of the space is assigned some other set called its closure and this assignment is called the closure operation. Now we say that a point is close to a set ‘  iff is in the closure of . Note that the closure operator has to satisfy some properties like , . These properties give a closeness relation which satisfies : (i) If a point is close to set of points close to A, then it is also close to A.(ii) If a point is close to a union of two subsets, then it must be close to at least one of them.(iii) No point is close to .

So a topological space may be viewed as set with a notion of closeness between points and subsets, and continuous functions as just the maps preserving this additional structure of sets. And this, for me, looks a more natural way to think of continuity and topological spaces.