On the dimension of ordered spaces
- Autores: Bernard Brunet
- Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 48, Fasc. 3, 1997, págs. 303-314
- Idioma: inglés
- Enlaces
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Resumen
In this paper, we prove that, for every ordered space (ordered set with its order topology i.e. with the topology generated by the family of all intervals $]\leftarrow, a[$ and $]a,\rightarrow [)$ and more generally for every line (space homeomorphic to a subspace of an ordered space and called in (7) generalized ordered space), the small inductive dimension (\textit{ind}), the large inductive dimension (\textit{Ind}), the covering dimension (\textit{dim}) and the nonstandard definition or thickness (\textit{ep}) coincide. More precisely, we prove, that for every line $X\not=\emptyset$, we have: \begin{enumerate}[1.)] \item $ep X = ind X = Ind X = dim X = 0$ if and only if $X$ is totally disconnected. \item $ep X = ind X = Ind X = dim X = 1$ if and only if $X$ is not totally disconnected\end{enumerate}.
