Figure 8 and Circle Homeo?

May 8, 2010

I’m guilty of continuously mixing up homeomorphisms and homotopy equivalences.  It’s terrible, really.

So, in honor of this, let’s show that the circle and the figure 8 both viewed as subsets of {\mathbb R}^2 are not homeomorphic.  This is the same sort of deal as the plane verses line example in my last post, and there’s a really common trick when considering homeomorphisms: take out a point.

So, let’s take out a point of the circle and a point of the figure 8.  There’s actually a few cases to consider, since the figure 8 has two “kinds” of points on it.  But when we take a point out of a circle, what do we get?  We get a connected open line segment — or, at least, we get something homeomorphic to an open line segment.

Say we take out the “center point” of the figure 8.  What do we get?  We get two “cut” circles which are not attached to each other, and these are homeomorphic to two disjoint open line segments.  Since \pi_0 (the number of components) differs in these, they must not be homeomorphic (see the end of the last post if you don’t understand why).

But what if we take out a point that’s not a center point of the figure 8?  We get a circle with horns — something that kind of looks like the Taurus astrology sign.  So, we’ve reduced this to proving that the circle and the line are not homeomorphic.  This is an exercise for the reader.

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