What is the topologist’s sine curve? Why is this curve attributed to topologists? If you Google Topologist’s Sine Curve, Evelyn Lamb’s article pops up, which does an excellent job of explaining the intuitive idea behind the topologist’s sine curve and why it is an interesting object to mathematicians (this is the link). This is an article she wrote for Scientific American. I chanced upon the article recently. That reminded me of an article I wrote on the topologist’s sine curve in my topology blog (my article).
I will not try to explain too much here, except to say that the topologist’s sine curve is good example of a connected space that is not path connected. Unlike many descriptions of the topologist’s sine curve, the topologist’s sine curve in my article does not use the function . My topologist’s sine curve is boxy. There are actually more than one curve. The following are two diagrams from my article.
Figure 3 is the closed topologist’s sine curve (closed because it has the vertical bar at the left). This curve is identical (topologically speaking) to the plus the vertical bar at the left. In my opinion, the boxy version is better in some way. This curve is constructed by an iterative process (please see Figure 1 and Figure 2 in my article). As a result, the boxy curve brings out the essential idea more clearly – the curve is connected and yet not path connected. The whole curve is connected. Yet you cannot get from the right edge to the left edge.
Figure 4 is the extended topologist’s sine curve, which has an additional bar at the bottom. The bar at the bottom makes it path connected. You can now go from any point in the curve to the vertical bar at the left side.
Please feel free to read any or all of these articles.
Evelyn Lamb’s article also has links to other pieces that she wrote on other interesting mathematical objects (Mobius strip is one).