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….

__Exercise 1.2__

**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^{[1]}

and

.

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 ^{[2]}. 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.

__Exercise 1.3__

**Show that is compact. [Hint: Show that the restriction of to is surjective.]**

*Proof.*

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^{[3]} 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.

2. The first set description displayed was borrowed from here. The second set description was mine, but I liked the first better so I included it for completeness.

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 .