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 "nondegenerate" any set of points containing more than one point. Now every example in the book of a nondegenerate continuum contains at least one proper nondegenerate subcontinuum. But nowhere in the book is there a theorem stating that if C is a nondegenerate continuum, then C always contains a proper nondegenerate subcontinuum (unless further restrictions are placed on Csuch as being locally compact). Does such an "unrestricted theorem" actually not follow from the Moore space axioms.?
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Here is a partial answer. There is a space (KnasterKuratowski fan) which is a subset of RxR, and which is connected but becomes totally disconnected upon removal of one point. There may be extensions of this example in which removal of any point causes the extension to be disconnected. I do not know of (but can posit the existence of) a space in which removal of any point causes the subspace to be totally disconnected. This might then be a candidate for a Moore space with no proper nondegenerate Moore spaces. Gerhard "Ask Me About System Design" Paseman, 2010.07.28 

