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

Mathematics Asked by Saikat Goswami on August 25, 2020

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.

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

## Related Questions

### (proposed) elegant solution to IMO 2003 P1

0  Asked on January 18, 2021 by mnishaurya

### Reverse order of polynomial coefficients of type $left(r-xright)^n$

1  Asked on January 18, 2021 by thinkingeye

### Should one include already cemented proofs of related principles in one’s paper?

0  Asked on January 18, 2021 by a-kvle

### The Cauchy-Crofton formula on a plane

1  Asked on January 18, 2021 by jalede-jale-uff-ne-jale

### Solving a limit for capacity of a transmission system

3  Asked on January 18, 2021 by jeongbyulji

### Galois group Abstract algebra

1  Asked on January 18, 2021 by user462999

### Does there always exists coefficients $c,dinmathbb{R}$ s.t. $ax^3+bx^2+cx+d$ has three different real roots?

2  Asked on January 17, 2021 by w2s

### If there is a “worldly ordinal,” then must there be a worldly cardinal?

2  Asked on January 17, 2021 by jesse-elliott

### Cover number and matching number in hypergraphs.

1  Asked on January 17, 2021 by josh-ng

### $displaystyleint_C (e^x+cos(x)+2y),dx+(2x-frac{y^2}{3}),dy$ in an ellipse

1  Asked on January 17, 2021 by fabrizio-gambeln

### Show that $h_n(x)=x^{1+frac{1}{2n-1}}$ converges uniformly on $[-1, 1]$.

1  Asked on January 17, 2021 by wiza

### What does the functional monotone class theorem say and how does it relate to the other monotone class theorem?

0  Asked on January 17, 2021 by alan-simonin

### Symmetric matrix and Hermitian matrix, unitarily diagonalizable

2  Asked on January 17, 2021

### direct conversion from az/el to ecliptic coordinates

0  Asked on January 16, 2021 by klapaucius-klapaucius

### A proof problem in mathematical statistics

2  Asked on January 16, 2021

### How to find the z component of the parameterization of an ellipse that is the intersection of a vertical cylinder and a plane

1  Asked on January 16, 2021 by nono4271

### using Lagrange multipliers to determine shortest distance between a point and straight line

2  Asked on January 16, 2021 by am_11235

### Proof of Lemma 5.1.5.3 in Jacob Lurie’s HTT.

1  Asked on January 16, 2021 by robin-carlier