A Method for Constructing Ordered Continua

In recent years some types of topological spaces were constructed having only the ‘necessary” continuous self-maps (of a special kind). Examples of this type are: a topological group having no other continuous self-maps other than the translations and the constant maps [5] and an infinite-dimensional inner-product space with only trivial bounded linear operators (an operator A is trivial if for some scalar A, A-AI has finite dimensional range) [6]. For older results of this type see [3, 71. We pursue this line a little further by constructing an ordered continuum with only the necessary continuous self-maps: for an explanation of ‘necessary’ in this context see Section 5. Actually our main result is a general method for constructing ordered continua, of which the above-mentioned continuum is an illustration. A second example is presented in Section 4 (this example came first in time), which is an order-homogeneous non-reversible ordered continuum. The first (real) example of this type was constructed by Shelah [S]. Our example is totally different from Shelah’s (see Section 4 for an explanation) and, in our opinion, somewhat simpler. The paper is organized as follows: Section 1 contains the necessary definitions and preliminaries. Section 2 concerns special subsets of the unit interval [0, 11. The construction presented there is very much like the one in [4]. In Section 3 we show how to construct ordered continua from families of subsets of [0, I]. In Sections 4 and 5 we construct the continua mentioned above using the method of Section 3, with input from Section 2.