Relation between IVT, Connectedness, Completness in Metric Space, LUB

Question

The motive behind this question is:

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

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.

metric spaces real analysis

Brian M. Scott · Answer

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.)

Answered by Brian M. Scott on August 25, 2020

Relation between IVT, Connectedness, Completness in Metric Space, LUB

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