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I know that a the underlying set of a metric space is uncountable whenever the metric space is connected and the underlying set contains at least two points. In A connected set with at least two points has no isolated points, it has also been shown that this not true if we generalize the metric space to a non-metrizable connected topological space where the underlying set doesn't even need to be infinte.

Now I'm trying to figure out if any such relations exist for the cardinality of the underlying sets (having at least two members) of Connected or Compact topological ( but not necessarily matrizable) spaces? I also have the same question regarding the relation between the cardinality of the underlying sets of metric spaces and the space being complete (but not necessarily compact)?

To make myself clear I'm seeking for such relationships dealing with Connectedness and Compactness purely topologically (i.e. forgetting that a metric space can also be a topological space) and dealing with completeness completely analytical (i.e. forgetting again that a metric space can also be a topological space)?

sanaz mat
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  • I don't have one. I believe completeness is a property of metric spaces and can not be defined just by topology and clearly the fact that completeness is not necessarily preserved by homeomorphisms shows this. – sanaz mat Jan 10 '18 at 14:10

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Any nonempty set with the trivial topology (i.e., the only open sets are the empty set and the full space) is both compact and connected.

For completeness, any nonempty set with the discrete metric (i.e., $d(x,y) = 1\;$if$\;x \ne y$) is a complete metric space.

quasi
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