What continuity really is (pretentious heading to attract readership)

by ayushkhaitan3437

This is an article I wanted to write for a long time.

What exactly is topological continuity? A mapping f:X\to Y between two topological spaces is said to be continuous if for any open set U\subset Y, f^{-1}(U) is open in X.

After reading this definition, very few students understand the motivation behind this definition. By bringing these concepts to Euclidean space, a  picture slowly begins to form. This definition is valid in \Bbb{R^n}. However, what was the need to unnecessarily generalize continuity to non-Euclidean spaces? What does continuity really mean in a general setting?

Given below is my impression of continuity, and has not been influenced by any mathematical work that I have come across.

If there is a continuous mapping from X to Y, this means that Y is at least as “solid and smooth”, or more “solid and smooth” than X. I have no intention of making the phrase “solid and smooth” precise, and it is an arbitrary phrase concocted by me to give readers a “feel” for the nature of continuity. As an intuitive explanation, a circle is more “solid and smooth” then a line, which has two sharp points. The graph of y=x is more “solid and smooth” than y=\lfloor x\rfloor. Developing more examples of this kind would be a useful exercise. The less the number of “sharp points”, the more an object is “solid and smooth”.

Now we swoop back to my original claim: that if f:X\to Y is continuous, then Y is at least as or more “solid and smooth” than X. As an example, consider the continuous mapping of [0,1] to \Bbb{S}^1. The continuous map for this would be t\to (\cos 2\pi t,\sin 2\pi t). We know that [0,1] has two “sharp points” (at 0 and 1), while \Bbb{S^1} has none. Hence, \Bbb{S}^1 is at least as or more solid and smooth than [0,1]. However, there is no continuous mapping from \Bbb{S}^1 to [0,1], as the latter is less solid and smooth than the former.

It is a useful exercise to use this concept in more complicated topological cases.

Now we come to path homotopy. When we say there is a continuous mapping from [0,1]\times [0,1] to a topological space such that f(I\times \{0\})=C_1 and f(I\times \{1\})=C_2, then we are in essence saying that the space encompassed and bordered by C_1 and C_2 is as or more solid and smooth then I_0^2. Hence, there are no breaks in between. It can be pictured as a solid block of ink. No spaces.

This is the reason why [0,1] is generally used as the domain for a continuous map. We want the image to be solid. No gaps.