# A question about Moore spaces.

Moore in his classic book "Foundations of Point Set Theory" defines a "continuum" to be a set of points that is both closed and connected (but not necessarily compact). He also calls "non-degenerate" any set of points containing more than one point. Now every example in the book of a non-degenerate continuum contains at least one proper non-degenerate sub-continuum. But nowhere in the book is there a theorem stating that if C is a non-degenerate continuum, then C always contains a proper non-degenerate sub-continuum (unless further restrictions are placed on C-such as being locally compact). Does such an "unrestricted theorem" actually not follow from the Moore space axioms.?

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You could use a title that asks the whole question: Does every non-degenerate continuum contain a proper non-degenerate sub-continuum? –  Joel David Hamkins Jul 28 '10 at 19:03
Since "closed" is a relative notion, it sounds like Moore's continua are subsets of some ambient topological space. Maybe Euclidean $n$-space for some $n$? Please clarify. –  Pete L. Clark Jul 28 '10 at 19:38
The title may be a bit confusing: what is usually called a Moore space is a CW complex whose reduced homology in given degree is isomorphic to a given abelian group and is zero in other degrees. –  algori Jul 28 '10 at 21:29
The title should be changed; "Moore space" has a specific meaning different from what you intend... –  Romeo Nov 4 '10 at 6:54