I’ve always meant to come back to these at some point.
Ex7
Compact subsets of the plane are bounded (cover such a set by an expanding sequence of open balls and apply compactness). This means we can view S as a subset of not just R2 but also the closed triangle T⊂R2, appropriately scaled. Let π:T→S be the projection (the exercise tells us that π is continuous (compact subsets of Hausdorff spaces are closed) ). Then f∘π:T→S is continuous and since S⊆T, it has a fixed point x∗ according to Ex6. Since the image of f∘π is just S, this fixed point must lie in S, and since f∘π≡f on S, it’s the desired fixed point for f.
I’ve always meant to come back to these at some point.
Ex7
Compact subsets of the plane are bounded (cover such a set by an expanding sequence of open balls and apply compactness). This means we can view S as a subset of not just R2 but also the closed triangle T⊂R2, appropriately scaled. Let π:T→S be the projection (the exercise tells us that π is continuous (compact subsets of Hausdorff spaces are closed) ). Then f∘π:T→S is continuous and since S⊆T, it has a fixed point x∗ according to Ex6. Since the image of f∘π is just S, this fixed point must lie in S, and since f∘π≡f on S, it’s the desired fixed point for f.