Research and Exposition in Mathematics -- Volume 13
Enlarged Picture
W. Taylor
The Clone of a Topological Space
100 p., soft cover, ISBN 3-88538-213-X, EUR 20.00, 1986
The set of all continuous selfmaps of a topological space A forms a semigroup,
whose properties are well known to reflect those of A in interesting ways. Here
we will consider a more general structure, called the clone of A, which consists
of all continuous maps from the product of n copies of A to A,
n ranging over all integers.
Several theorems are proved indicating how many basic topological properties of A
(connectedness, fixed point properties, dimension, etc.) are equivalent to some corresponding
algebraic or first order properties of the clone C(A). Most interesting results are
obtained for homotopy properties, and for particular spaces A, C(A) can be
interpreted as a homotopy group.
Finally this approach is related to the well known theory of the ring C(A) of all
continuous real-valued functions from C(A) to R and it is shown that this
ring can be interpreted in the clone C(A).