The Freudenthal space for approximate systems of compacta and some applications

In this paper we define a space $\sigma (\underline{\bold X})$ for approximate systems of compact spaces. The construction is due to H. Freudenthal for usual inverse sequences \cite{4, p. 153--156}. We stablish the following properties of this space: (1) The space $\sigma(\underline{\bold X})$ is a paracompact space, (2) Moreover, if $\underline{\bold X}$ is an approximate sequence of compact (metric) spaces, then $\sigma(\underline{\bold X})$ is a compact (metric) space (Lemma 2.4). We give the following applications of the space $\sigma(\underline{\bold X})$: (3) If $\underline{\bold X}$ is an approximate system of continua, then $X=\lim \underline{\bold X}$ is a continuum (Theorem 3.1), (4) If $\underline{\bold X}$ is an approximate system of hereditarily unicoherent spaces, then $X=\lim \underline{\bold X}$ is hereditarily unicoherent (Theorem 3.6), (5) If $\underline{\bold X}$ is an approximate system of trees with monotone onto bonding mappings, then $X=\lim\underline{\bold X}$ is a tree (Theorem 3.13).