These are exercises from the first few pages of Lee’s Introduction to Smooth Manifolds. These are pretty elementary from a topology perspective – maybe the middle-to-late part of a semester on point-set topology – but I decided to put them here to (a) remember some of that stuff, and (b) force myself to not delay doing them any longer. As it turns out, I struggled more with these than I should have and even had to consult the internet for a couple thing; in instances of internet borrowing, I provide links.
Note, finally, that if you’re trying to hunt these particular exercises down within Dr. Lee’s book, they’re actually dispersed throughout the text; suffice it to say, I’m not a fan of that particular method of delivery. Nevertheless….
Show that is Hausdorff and second countable and is therefore a topological -manifold.
Proof. Note that proving Hausdorffness and second countability will exactly prove the -manifold claim due to the discussion in Example 1.3. Also recall that by Lee’s definition, is defined to be the collection of 1-dimensional linear subspaces of with the quotient topology determined by which sends every point to the 1-D subspace spanned by . Perhaps a more intuitively useful definition is that is the space obtained from with the identification for all . That’s the construction used hereon.
To prove Hausdorffness, suppose are distinct points in and consider the subspaces and in . As a subspace of a metric space , is Hausdorff, whereby there exist open sets and in for which . Similarly for and , yielding open sets and which are also disjoint. For notational convenience, let denote the antipodal map, and consider the open sets
Openness and disjointness of the sets are obvious, and it’s easily verified that for . Hence, and are disjoint sets which are open (as insured by the fact that and are open in along with properties of the quotient topology) and which contain and , respectively.
To show second countability, note first that has a countable basis consisting of balls of rational radii with rational centers. Next, notice that the -sphere also has an induced countable basis whose elements are of the form . Finally, the claim is that the collection
is a basis for . Clearly, covers . Suppose, then, that , and consider an element where , . Because is a basis for , there exists an element containing (respectively an element containing ) for which (respectively, for which ). Thus, as claimed, is a basis for in one-to-one correspondence with the basis of and hence is a countable basis.
Show that is compact. [Hint: Show that the restriction of to is surjective.]
Clearly, is surjective due to the fact that for given , where . It’s also easily shown that is an open map, whereby we have that is a quotient map. One well known property of quotient maps is that if the host space is compact, so too are all its quotients; in particular, then, because is compact, – that is, the image of under the quotient map – is also compact. Hence, the result.
1. I was at first oblivious to the need for unions and intersections. That bit – and therefore the subsequent part – comes from here.
3. To prove it: Let be an open cover of . Because is open in , is open in . Moreover, is an open cover of , and because is compact, there exists a finite subcover of . But then is a finite subcollection of which covers .