On Topological Inverse Semigroup of Finite Transformations of an Infinite Set

We establish topological properties of the topological symmetric inverse semigroup of finite transformations I n λ of the rank 6 n. We show that the topological inverse semigroup I n λ is algebraically closed in the class of topological inverse semigroups. Many topologists established topological property of topological spaces of partial continuous maps PC (X, Y ) from a topological space X into a topological space Y with various topologies. For example: Vietoris topology, generalized compact-open topology, graph topology, τ -topology, and other (see [1, 7, 10, 13, 19, 20, 21, 22]). Since the composition of partially transformations of a topological space X is an associative operation, the set of all partial continuous transformations PC T (X) of the space X with the operation composition is a semigroup. Therefore many semigrouppers established the semigroup of continuous transformations of a topological space (see surveys [23] and [15]). Also many authors established semigroups of partial homeomorphisms of arbitrary topological space (see [2, 3, 4, 5, 14, 24, 29, 34]). Bĕıda [6], Orlov [25, 26], and Subbiah [32] established semigroup and semigroup inverse topologies of semigroups of some classes partial homeomorphisms of some classes of topological spaces. In our paper we obtained a very curious result: if we consider an inverse semigroup of partial finite bijection I n λ of the rank 6 n of the discrete topological space (and hence any Hausdorff topological space) with any semigroup inverse topology τ , then (I n λ , τ) is a closed subsemigroup of any topological semigroup which contains I n λ as a subsemigroup. In this paper all topological spaces will be assumed to be Hausdorff. We shall follow the terminology of [8, 9, 12, 27, 28]. If A is a subset of a topological space X, then we denote the closure of the set A in X by clX(A). A semigroup S is called an inverse semigroup if every a in S possesses an unique inverse, i.e. if there exists an unique element a in S such that aaa = a and aaa = a. A map which associates to any element of an inverse semigroup its inverse is called the inversion. A topological (inverse) semigroup is a topological space together with a continuous multiplication (and an inversion, respectively). Obviously, the inversion defined on a topological inverse semigroup is a homeomorphism. If S is a semigroup (an inverse semigroup) and τ is a topology on S such that (S, τ) is a topological (inverse) semigroup, then we shall call τ a semigroup (inverse) topology on S. If S is a semigroup, then by E(S) we denote the band (the subset of all idempotents) of S. On the set of idempotents E(S) there exists a natural partial order: e 6 f if and only if ef = fe = e. If A is a subset of an inverse semigroups S, then we denote A = {x | x ∈ A}. By ω we denote the first infinite ordinal. Further, we identify all cardinals with their corresponding initial ordinals. Let X be a set of cardinality λ > 1. Without loss of generality we can identify the set X with the cardinal λ. A function α mapping a subset Y of X into X is called a partial transformation Date: September 29, 2008. 2000 Mathematics Subject Classification. Primary 22A15, 20M20. Secondary 20M18, 54H15.