H-problems

This list is maintained by Jeremy Smith, whom you can contact by jeremy@math.ufl.edu via email.

Convention: A boldface x to denote the sequence ( x₁, x₂, x₃, ... ) and similarly for y and z.

H-100:   12Jan1998

Let T be the topology on the real line consisting of all sets which are of one of the following forms: R (the entire real line), the empty set, or the interval (y,infinity), where y is a real number.

1. Give a sufficient condition for a sequence x to converge to a real number 'b'.
2. Give a necessary condition for a sequence x to converge to a real number 'b'.
3. Fix a sequence x of real numbers. Define L(x) to be the set of real numbers b such that       lim x   =   b.

   (i)

   Can L(x) be a closed interval? In particular, can L(x)=(-infinity,0]?

   (ii)

   Can L(x) be an open interval? In particular, can L(x)=(-infinity,0)?

   (iii)

   Can L(x) be the empty set?

H-101:   14Jan1998

Answer all of the same questions if T is generated by the basis/subbasis S={[y, infinity): y a real number} along with the entire real line and the empty set.

H-102:   18Jan1998

Is it the case that if a space X is not sequentially compact then there exists an open cover with no Lebesgue number?

H-103:   21Jan1998

Find a characterization of the property "C is a connected subset of X" without directly using the induced topology on C.

H-104:   28Jan1998

Let W be a chain with the Order Topology. Show that W is regular.

H-105:   28Jan1998

We showed in H-104 that each chain (under the order topology) is regular. How much can this be strengthened? In particular, letting 'Omega' denote the first uncountable ordinal, is Omega a normal TS?

H-106   03Feb1998

In class we proved the following Lemma: If (W,<) is a chain and y is a sequence contained in W, then there exists a subsequence of y which is monotonic. Find an interesting generalization of this Lemma.

H-107   09FEB1998

Prove or disprove the following statement: If a TS X is locally compact and Hausdorff, then X is normal. (Note that if X is Compact, then X is normal.)

H-108   09Feb1998

Suppose X is a Hausdorff TS, and A,B,C are compact subsets of X, and p is a point of X. Show that:

[1] A \inter B is compact.

[2] C closure is compact.

[3] p has a compact neighborhood IFF there is an open set U owning p such that U closure is compact.

[4] Now assume that X is locally compact at p. Prove that inside each open neighborhood V of p, there is a compact neighborhood of p.

H-109   02Mar1998

Equip the integers, Z, with the co-finite topology.

(a) Show that Z is connected and locally connected

(b) Show that Z is NOT locally path-connected. Show that the path-component of a path n is simply (n).

H-110   02Feb1998

Put the usual metric, call it u, on the interval I := (-3,3). Put the corresponding uniform metric, call it ubar, on the countable product Lambda := I x I x I x ....

Is (Lambda, ubar) connected? What is the connected component of the point hat0 := (0,0,0....)?

H-111   28Mar1998

Show that the family of Lower Semi Continuous functions is sealed under supremum.

H-112   28Mar1998

Is I:=((-3,3)^omega, Uniform Metric) connected?

H-113   28Mar1998

Let Omega be a topological space. Recall that one compactification, Y, of Omega is said to be under another compactification, A, if there exists a continuous function phi:Y-->A such that phi post-composed with the identity map on [TO BE CONTINUED]...Suppose that Omega is locally compact and Hausdorff, but not compact. Let A be the one-point-compactification of Omega. Show that this compactification is 'under' every other compactification Y by a map phi:Y-->A defined by phi(w)=w for w in Omega, phi(y)=infinity for y in Y-Omega.

H-115  12Apr1998

Quexercise: Let Omega be a complete metric space. Suppose that there exists a sequence of nonempty closed subsets of Omega, say (C_n)₁^Infty, that is nested 'going up' [i.e., C_n is included in C_n+1]. Furthermore, suppose that the diameter of C₁ is finite. Must the intersection of the C_n's be nonempty?

H-116 12Apr1998

Let M_2x2(R) denote the set of 2x2 matrices with entries from R. Let "E" denote the subset of M_2x2 consisting of those matrices with nonzero determinant. Show that E is arcwise connected. Hint: Fix a matrix A in E. Connect A to the 2x2 Identity matrix. Extend this argument to the general nxn case.