Let be a Moufang plane. By specializing one line ω, the line at infinity, weobtain an affine Moufang plane . The group generated by the shears of is called the equiaffine group. Veblen [9, § 52] asked whether every equiaffinity is a product of two affine reflections. He gave a proof which will work in an affine Pappian plane, using the following two properties.

Property 1. If an equiaffinity fixes two distinct proper points of , it fixes every point collinear with them.

Property 2. Let e be an equiaffinity and P a point such that ppe2pe3 is a triangle. Then the lines PePe2 and PPe3 are parallel.

Without using these properties, it will be proved that the answer to Veblen's question is “yes“ if and only if the Moufang plane is Pappian.

Let X(t) denote real-valued Brownian motion on the interval 0 ≦ t ≦ 1, so normalized that E(X2(t)) = t. We prove some theorems about transforms X(F) of closed sets F: in general, F is not known in advance but depends on X. The main point of comparison among sets is taken to be their Hausdorff dimension, and in this respect the linear process is quite different from the planar. We state and discuss briefly two theorems.

(A) It is almost sure that, for every closed set F in [0, 1],

(B) For each closed set F in (—00,00) and number a ,

Plainly, statements (A) and (B) are nearly best possible. For the planar process dimX(F) = 2 dim F (with the same quantification as in (A)) [6].

In [1] de Groot has introduced the notation “connectedly generated” (or eg) for those spaces in which the closed connected sets form a subbase for the topology. He pointed out that these are the semi-locally connected spaces of Whyburn. See [5; 6].

If X is cg, then, since X is closed, X is the union of a finite number of closed connected sets and, thus, has only a finite number of components. If p is any point in a eg space, and Nv is any neighbourhood of p, then the complement of Nv may be covered by a finite number of closed connected sets, none of which contain p.

In [1] and in [2] the concept of “screening” is introduced and shown to be usefully related to local connectedness and construction of compactifications for completely regular spaces. We review this concept in § 2.

This paper is a sequel to [2], whose primary purposes are to clarify and generalize the concept introduced there of an eigenfunction of an inner function, and to answer questions raised there concerning the equivalence of several possible forms of the definition. A new definition, proposed here, leads to a complete characterization of the eigenfunctions of Potapov inner functions of normal operators, and the result is more satisfactory than [2, Theorem 3.4], although the latter is used strongly in the proof.

Let be an inner function in the sense of Lax; i.e., is almost everywhere (a.e.) a unitary operator on a separable Hilbert space and belongs weakly to the Hardy class H2. An analytic function q (which will have to be a scalar inner function) was defined to be an eigenfunction of if the set of z in the disk {z: |z| ≦ 1} for which is invertible is a set of linear measure 0 on the circle {z: |z| = 1}.

In recent years there has been some interest in trying to improve the behaviour of maps by extending their domains. For example, in 1953 Whyburn showed that every map is the restriction of a compact map [17]. Similarly, Krolevec proved in 1967 that each locally perfect map can be extended to a perfect map [12] and in an as yet unpublished paper, Dickman obtained the same result for arbitrary maps [4]. In this paper we show that every map can be extended to an open map so that certain properties of the domain and range are preserved in the new domain. These results are then used to obtain analogues and improvements of recent theorems of Arhangel'skiï, Ĉoban, Hodel, and Proizvolov.

In this paper we present three results involving function constructions, together with an example which shows the usefulness of our considerations. The first result, Theorem 1, states roughly that homotopy is self related through function space constructions. Section 1 is devoted to the precise statement and proof of this theorem.

In order to clarify the results of § 2 we draw an analogy. Recall that under certain mild restrictions on the spaces involved, a map p: E → B is a fibration, and a map i: A → X is a cofibration, if and only if the induced maps p*: EZ → BZ and i*: ZX → ZA are fibrations for all spaces Z. The two theorems of § 2 give, in a more general category, analogous results for the notions of limit and colimit. Finally, in § 3 we suggest the importance of this type of result by using it, together with Theorem 1 and the above analogy, to deduce a result from its dual.

In 1967 Foulser [1] defined a class of translation planes, called generalized André planes or λ-planes and discussed the associated autotopism collineation groups. While discussing these collineation groups he raised the following question:

“Are there collineations of a λ plane which move the axes but do not interchange them?”.

In this context, Foulser mentioned a conjecture of D. R. Hughes that among the André planes, only the Hall planes have collineations moving the axes without interchanging them. Wilke [4] answered Foulser's question partially by showing that the conjecture of Hughes is indeed correct. Recently, Foulser [2] has shown that possibly with a certain exception the Hall planes are the only generalized André planes which have collineations moving the axes without interchanging them. Our aim in this paper is to give an alternate proof, which is completely general, and is in the style of the original problem.