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The motive behind this question is:

  1. In an ordered field LUB <=> IVT. Is the IVT equivalent to completeness?

  2. Connected Metric Space X does have IVP i.e "all continuous function f:M⟶R that admit a positive value and a negative value, also admit a c∈M such that f(c)=0". I read that the converse is also true. If M a metric space with the property of Intermediate Value. Show that M is connected.

So LUB <=> IVT <=> Connectedness (Is this correct?)

  1. Does the Complete Metric Space also have this property?

I am trying to find is a connection between Completion in Metric Space and Connectedness in Metric Space.

Also, Cauchy Completeness in R is weaker than LUB in R.(Cauchy Completion + Archimedian Property $\implies$ LUB)

So does Completion in $\mathbb{R}$ is stronger than Completion in Metric Spaces?

Basically I am trying to see the connections between all the the properties of Real Analysis and Metric Spaces.

Saikat
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1 Answers1

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There is no connection between completeness and connectedness. The middle-thirds Cantor set is complete in the usual metric and is zero-dimensional and hence totally disconnected. The irrationals are not complete in the usual metric, but they are a $G_\delta$-subset of the complete metric space $\Bbb R$, so there is a metric on them that generates the usual topology and in which they are complete. (Every $G_\delta$-set in a complete metric space is completely metrizable.)

Brian M. Scott
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