- Show the intersection of any collection of closed sets is closed and the union of any collection of open sets is open.
- Show that if H is closed and U is open, then H ∖ U is closed. Next show that U ∖ H is open.
- Show the finite intersection of any collection of open sets is open.
- Show the finite union of any collection of closed sets is closed.
- Suppose
_{n=1}^{N}is a finite collection of sets and suppose x is a limit point of ∪_{n=1}^{N}H_{n}. Show x must be a limit point of at least one H_{n}. - Give an example of a set of closed sets whose union is not closed.
- Give an example of a set of open sets whose intersection is not open.
- Give an example of a set of open sets whose intersection is a closed interval.
- Give an example of a set of closed sets whose union is open.
- Give an example of a set of closed sets whose union is an open interval.
- Give an example of a set of open sets whose intersection is closed.
- Give an example of a set of open sets whose intersection is the natural numbers.
- Explain why F and ∅ are sets which are both open and closed when considered as subsets of F.
- Let A be a nonempty set of points and let A
^{′}denote the set of limit points of A. Show A ∪ A^{′}is closed. Hint: You must show the limit points of A ∪ A^{′}are in A ∪ A^{′}. - Let U be any open set in F. Show that every point of U is a limit point of U.
- Suppose is a sequence of sequentially compact nonempty sets which have the property that K
_{n}⊇ K_{n+1}for all n. Show there exists a point in the intersection of all these sets, denoted by ∩_{n=1}^{∞}K_{n}. - Now suppose is a sequence of sequentially compact nonempty sets which have the finite intersection property , every finite subset ofhas nonempty intersection. Show there exists a point in ∩
_{n=1}^{∞}K_{n}. - Show that any finite union of sequentially compact sets is compact.
- Start with the unit interval, I
_{0}≡. Delete the middle third open interval,resulting in the two closed intervals, I_{1}=∪. Next delete the middle third of each of these intervals resulting in I_{2}=∪∪∪and continue doing this forever. Show the intersection of all these I_{n}is nonempty. Letting P = ∩_{n=1}^{∞}I_{n}explain why every point of P is a limit point of P. Would the conclusion be any different if, instead of the middle third open interval, you took out an open interval of arbitrary length, each time leaving two closed intervals where there was one to begin with? This process produces something called the Cantor set. It is the basis for many pathological examples of unbelievably sick functions as well as being an essential ingredient in some extremely important theorems. - In Problem 19 in the case where the middle third is taken out, show the total
length of open intervals removed equals 1. Thus what is left is very “short”. For
your information, the Cantor set is uncountable. In addition, it can be shown there
exists a function which maps the Cantor set onto , for example, although you could replacewith the square×or more generally, any compact metric space, something you may study later.
- Show that there exists an onto map from the Cantor set P just described onto
. Show that this is so even if you do not always take out the middle third, but instead an open interval of arbitrary length, leaving two closed intervals in place of one. It turns out that all of these Cantor sets are topologically the same. Hint: Base your argument on the nested interval lemma. This will yield ideas which go somewhere.
- Suppose is a sequence of sets with the property that for every point x, there exists r > 0 such that Bintersects only finitely many of the H
_{n}. Such a collection of sets is called locally finite. Show that if the sets are all closed in addition to being locally finite, then the union of all these sets is also closed. This concept of local finiteness is of great significance although it will not be pursued further here. - Show every closed and bounded subset of F is compact. Hint: You might first
show every set of the form + iis compact by considering sequences of nested intervals in bothand.
- Show a set, K is compact if and only if whenever K ⊆∪ℬ where ℬ is a set whose
elements are open balls, it follows there are finitely many of these sets, B
_{1},,B_{m}such thatIn words, every open cover of open balls admits a finite subcover.

- Show every sequentially compact set in ℂ is a closed subset of some rectangle of the
form ×. From Problem 23, what does this say about sequentially compact sets being compact? Explain.
- Now suppose K is a compact subset of F which means every open cover admits
a finite subcover. Show that K must be contained in some set of the form
+ i. When you have done this, show K must be sequentially compact. Hint: If the first part were not so,
_{n=1}^{∞}would be an open cover but, does it have a finite subcover? For the second part, you know K ⊆+ ifor some r. Now if_{n=1}^{∞}is a sequence which has no subsequence which converges to a point in K, you know from Proposition 4.7.15 and Theorem 23, since+ iis sequentially compact, there is a subsequence,_{k=1}^{∞}which converges to some x + iy ∈+ i. Suppose x + iyK and consider the open cover of K given by_{n=1}^{∞}whereYou need to verify the O

_{n}are open sets and that they are an open cover of K which admits no finite subcover. From this you get a contradiction. - Show that every uncountable set of points in F has a limit point. This is not necessarily true if you replace the word, uncountable with the word, infinite. Explain why.

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