Topology H-Problems		Fall 1997

This is the list of randomly-assigned spur-of-the-moment Homework
Problems for Prof. King's first semester topologyy course of Autumn 1997.
This list was maintained by Kathy Reiff.

Here is the list as of 17Dec1997:


H-PROBLEMS
----------

H-1:  	Show that a finite poset is determined by its links.

H-2:	Create a lattice in which some infinite subset has no least upper
	bound.

H-3:	What is the greatest lower bound and least upper bound relationship
	among the elements of the following lattices?  Which of the
	lattices are distributive?  If they are not, then give a specific
	triple showing why it is not.
	  (a)	( Z+ , | ) = (positive integers, divisibility)
	  (b) 	( P(X) , < ) = (power set of X, inclusion)
	  (c)
	            a
		   /|\
	          / | \
	         d  e  b
	          \ | /
	           \|/
		    c

	  (d)
		    a
		   / \
	          /   b
	         e    |
	          \   c
	           \ /
	            d

H-4:	Write a rigorous proof that gcd distributes over lcm AND lcm
	distributes over gcd.  That is, show that wedge and join distribute
	over one another in ( Z+ , | ).

H-5:	(a)  Write a specific bijection from the naturals to the integers.
	     (i.e.  find f: N <-> Z)
	(b)  Using the formula for the sum of the first n integers, write a
	     bijective formula f: NxN <-> N .  Also, give the formula for
	     the inverse of f.
	(c)  Show abstractly that a set B is not bijective with its power
	     set P(B).  (i.e.  Construct an element x in P(B) such that
	     f(b) does not equal x for any b in B.)

H-6:	Let A=ZxZ and let B=A rotated by d degrees (e.g. d=40 degrees).
	Using the Schroeder-Bernstein Theorem from class (not just the
	Axiom of Choice), show that there is a distance bounded bijection
	h: A -> B .  (i.e.  Show that there exists a D such that
	dist( h(a),a ) < D  for all a in A.)

H-7:	Prove the same result as in H6 only now let A and B be finite
	unions of lattices.

H-8:	Show that (A^B)^C is bijective with A^(BxC) by writing a
	"perfectly explicit" function.  (suggestion: use the 'natural'
	function)

H-9:	Let C:=C(R) be the set of continuous functions from R to R, where
	R represents the real numbers.  How big is C?
	(hint: the rationals are dense in R)

H-10:	Find an order preserving function from R to 2^N, where the order
	induced on R is < (the normal ordering) and the order on 2^N is
	strict containment.

H-11:  	(a)  Construct a set B conained in R (the real numbers) such that
	     B is bijective with (Omega)^3.
	(b)  Prove that for each countable ordinal 'a,' there exists a
	     subset A contained in R which is order isomorphic to 'a'.
	    (Hint:  Don't do induction--there is a 1 or 2 sentence proof
	       to (b). )

H-12:	Let X:={0,1}^N   (i.e. the 2-bit sequence indexed by the naturals).
	Given x,y in X define d(x,y):=1/n where n is the smallest natural
	number such that x(sub n) does not equal y(sub n).
	(a)  Prove that m is a metric.
	(b)  If 1/n is replaced by a(sub n), a sequence of numbers, what is
	     the condition on [a(sub n)], n=1 to infinity, so that m is a
	     metric?  (Preferrably show <=> )

H-13:	(a)  Prove that the sequence space X defined in H-12 is ultra-metric.
	(b)  Prove that the definition of "ultra metric" given by Kaplansky
	     (Set theory and Metric Spaces)  is equivalent to the following
	     statement:    Every triangle is isosceles AND the repeated length
	     (as determined by d) is greater than or equal to the length of
	     the third side.
	(c)  Give an example of a metric D and points a,b,c so that
		  dist(a,b) + dist(b,c) = dist(a,c)     [*]
	     holds for D.  For m:=D/(1+D), however, [*] fails on a,b,c.

H-14:	Assume the Marriage Lemma* is true for the finite case.  Use this
	to prove the infinite case:
	(a)  #[L(S)] >= #[S] ,  for all subsets S of the set of boys B,
	     where L(S):={girls who are linked to at least one boy}.
	(b)  Each boy likes finitely (i.e. compactly many) girls.

	* Marriage Lemma (finite case):
	  For a set B(boys) and G(girls) with B finite and part(a) holds,
	  there exists a marriage pattern such that all boys are married to
	  girls they like.

H-15:	Given [0,1] a subset of the reals, show that there exists a sequence
	{n(k)},k=(1 to infinity) such that either:
		x[n(1)] <= x[n(2)] <= ... <= ...  (non-decreasing sequence)
	   OR   x[n(1)] >= x[n(2)] >= ... >= ...  (non-increasing sequence).

H-16:   Form a bijection between R(the reals) and {0,1}^N.
	(Do not use one already seen in class!)

H-17:   Let I:={ (n1, n2,... ) | for all k such that nk is in Z>0  AND
			           n1 < n2 < ...}  .
	Note that I is contained in Z+^(Z+).  Prove that #(I)=c.
	(Hint: Apply Schroeder-Bernstein in a simple way.)

H-18:	Suppose X is a topological space and let B be a basis for its
	topology.  Call a subcollection C\subset B  a "B-cover" (of X), if
	the union of the elements of C is all of X.
	  PROVE: Fixing B, suppose that each B-cover has a finite subcover.
	         Then X is compact.

H-19:	Let X be a metric space and S a subset of P(X).
	(a)  Define B[sub S] to be the collection of all finite intersections
	     of elements of S.  Show that B(sub S) is always a basis.
	(b)  Show that the intersection of any family of any family of
	     toppologies on X is a topology on X.
	(c)  Let T be a topology generated by S.  Prove that T is equal to
	     the intersection of all M such that M is a topology on X *and*
	     M contains S.

H-20:	Let A be a countable (indexing) set and consider the set of
	topological spaces  X:= { ( X[sub a], T[sub a] ) | a in A }.
	Suppose that for each 'a', there exists a base B[sub a] for a
	topology T[sub a].
	DEFINE:
	  Base[sub H]:= {G | U[sub a(sub i)] is included in B[sub a(sub i)]}
	  Base[sub P]:= {G | G = Product(over i) of U[sub a(sub i)]}
   	PROVE:

H-21:   Exhibit an open cover C of the reals, so that it has no Lebesgue
	number.  [I.e, for each positive delta, there exists a delta-ball
	which sticks out of each C-patch.]

H-22:   (X,m) is a MS.  Prove that (Total-boundedness & completeness)
	implies that X is sequentially compact.
  	[Hint: Consider a sequence \vec{x}.  By TB, for each n the space can
	be covered by finitely many balls of radius 1/n.  Do a Cantor
	Diagonalization argument (similar to our proof that if 4 colors
	suffice to legally color every FINITE planar graph, then 4 colors
	suffice to color any DENUMERABLE planar graph) to show that some
	subsequence of \vec{x} is Cauchy.]

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