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Say D^ is a closed disk standardly-embedded in the plane, and D is its interior.

I have an argument to the effect that any homeomorphism h:D^-->D^ sends boundary pts

to boundary points, but it seems kind of clunky, and I wonder if someone has a nicer,

cleaner one:

First, we show interior points are sent to interior points:

We take x in D. Since h is a homeo., h(D) is open, and h(x) lies in h(D). By openness

of h(D), there is an open set Ux containing h(x), so interior points are sent to interior points.

It follows, by considering the exterior of D^ as the complement of the disk, that

exterior points (i.e., those in R^2-D^ )

Now, the kind-of-clunky part:

We show that if x is a boundary point of D , so that every 'hood Vx of x intersects

points both in the interior of D and in the exterior of D, the same is the case for

h(Vx).

So the points in Ext(D^)/\Vx are sent to points in Ext(D^)/\h(Vx) , as a consequence

of the fact that exterior points are sent to exterior points.

Anyone know a nicer way?

Same for the interior points.

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# Homeomorphisms Send Bdries to Bdries

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