Topological continuity is a key concept within mathematical theory. Topological analysis has opened paths to a deeper understanding of continuity at a more abstract level. However, topological continuity isn’t the only form of continuity mathematicians study—discrete continuity is one type of continuity that is separate from topological continuity.
Algebraic General Topology is one way of bringing these different concepts of continuity under a single heading. Incidentally, AGT also describes continuity in the same way that is described to speak of discrete continuity.
We’ll need to look at some crucial concepts within topology and then apply these to certain sets to see how the general definition of topological continuity applies to specific sets. Or, see how we can generalize the definition of topological continuity from instances of continuous functions defined between any two spaces.
What is a Topological Space?
A topological space is a set X together with a topology T on it—that’s the basic definition of a topological space.
Before I go into greater detail about topological spaces, it’s important that I discuss what a topology is.
Definition of a Topology
A topology T on a nonempty set X is a collection of subsets of X, called open sets, such that:
- The empty set and the set X are open.
- The union of an arbitrary collection of open subsets of X is open.
- The intersection of a finite collection of open subsets of X is open.
Speaking in general terms, the set X contains a topology T if X also contains collections of open sets that satisfy the above conditions. T is basically the subset of X which conforms to the above mentioned properties and if you can find sets like T in X, then X is a topological space.
The difference between two topological spaces depends on how we define open sets within the set. For example, we define open sets on R like (0,1)—this set contains all the values between 0 and 1, but not 0 and 1 themselves but this only applies to the real line and not other sets. A general topological definition of an open set is:
A set S, such that every point in S has a neighborhood contained in S
Consider the open set (0,1) in R. I can define a neighborhood around every point that is contained within it. The entire set of real numbers, incidentally, also qualifies as a topology.
Before I define topological continuity, I should mention that you can only define continuous functions between two topological spaces. The definition will make this clear as well:
A function f: S→T between two topological spaces is continuous if the pre-image f−1 (Q) of every open set Q⊂T is an open subset of S.
As a function, f will map open sets of S onto T. If f happens to map all open sets in S specifically onto open sets in T, then f is considered continuous.
Discrete Topologies and Continuity
Discrete topologies are ones where every individual point represents an open set and is also a closed set. The set of integers on the real line, for example, is a discrete topology.
Continuity on Discrete Topologies
A continuous function from a discrete topology to another topological space—let’s say the real line—would be one that maps each member of the discrete topology onto an open set.
I felt that it was important to discuss continuity on discrete topologies to distinguish this idea that of discrete continuity. There are fundamental differences between continuity in discrete topologies and discrete continuity—discrete continuity is defined as:
a function f (on a set U) is a continuous function from μ to ν iff f∘μ ⊆ ν∘f
Where μ and ν are directed graphs.
If you’re interested in topological spaces or continuous functions defined between different topological spaces and generalizations , you should read Algebraic General Topology Volume 1.
The book introduces my mathematical theory that generalizes limits across arbitrary discontinuous functions. As always, I’m open to new ideas and thoughts on my work and welcome open debate on any issue you find.
Learn more about pointless topology here.