What continuity really is (pretentious heading to attract readership)
This is an article I wanted to write for a long time.
What exactly is topological continuity? A mapping between two topological spaces is said to be continuous if for any open set , is open in .
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 . 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 to , this means that is at least as “solid and smooth”, or more “solid and smooth” than . 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 is more “solid and smooth” than . 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 is continuous, then is at least as or more “solid and smooth” than . As an example, consider the continuous mapping of to . The continuous map for this would be . We know that has two “sharp points” (at 0 and 1), while has none. Hence, is at least as or more solid and smooth than . However, there is no continuous mapping from to , 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 to a topological space such that and , then we are in essence saying that the space encompassed and bordered by and is as or more solid and smooth then . Hence, there are no breaks in between. It can be pictured as a solid block of ink. No spaces.
This is the reason why is generally used as the domain for a continuous map. We want the image to be solid. No gaps.