Title:

Tentmaps, twopoint sets, and the selfTietze property

This thesis discusses three distinct topics. A topological space X is said to be self Tietze if for every closed C eX, every continuous f: C + X admits a continuous extension F: X + X. We show that every disconnected, self Tietze space is ultranormal. The Tychonoff Plank is an example of a compact self Tietze space which is not completely normal, and we establish that a completely normal, zero dimensional, homogeneous space need not be self Tietze. A subset of the plane is a twopoint set if it meets every straight line in exactly two points. We show that a twopoint set cannot contain a dense G8 subset of an arc. We also show that the complement of a twopoint set is necessarily pathconnected. Finally, we construct a zerodimensional subset of the plane of which the complement is simplyconnected. For A E lR, the tentmap with slope A is the function f: [0, 1] + lR such that f(x) = AX for x :=:; ~ and f(x) = A(l  x) for x ~ ~. Properties of wlimit sets of tentmaps, i.e. sets of the form n {fn+k(x) I kEN} nEN for x E [0,1], are examined, and an example of a tentmap and a closed, invariant, nonempty, internally chain transitive subset of [0, 1] which is not an wlimit set is given.
