intervals are connected

We wish to show that intervals (with standard topology) are connectedPlanetmathPlanetmath. In order to this, we will prove that the space of real numbers is connected. First we need a lemma.

Let (X,d) be a metric space. Recall that for xX and r+ we have


Lemma. Let (X,d) be a metric space and R+ such that R is nonempty and boundedPlanetmathPlanetmathPlanetmathPlanetmath. Then for any xX we have


Proof. Assume that yrRB(x,r). Then there is r0R such that d(x,y)<r0 and thus d(x,y)<sup(R), so yB(x,sup(R)).

Now assume that yB(x,sup(R)). Then d(x,y)<sup(R) and it follows (from the definition of supremumMathworldPlanetmathPlanetmath) that there is r0R such that d(x,y)<r0 and therefore yB(x,r0)rRB(x,r), which completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

PropositionPlanetmathPlanetmathPlanetmath. The space of real numbers is connected.

Proof. Assume that U,V are open subsets of such that UV= and UV=. Furthermore assume that U and take any x0U. Then (since U is open) there is r0 such that the open ball


is contained in U. Consider the set


Thus R is nonempty.

Assume that R is bounded. Denote by s=sup(R)<. We can apply the lemma:


Thus (due to the definition of R) B(x0,s) is a maximal open ball (with the center in x0) which is contained in U. Now


for some a,b. Since (a,b) is maximal then aU or bU. Indeed, if both aU and bU, then (since U is open) small neighbourhoods of a and b are also contained in U, so (a-ϵ,b+ϵ) is contained in U (for some ϵ>0), but (a,b) was maximal. ContradictionMathworldPlanetmathPlanetmath.

Without loss of generality we can assume that bU. Then bV, because UV=. But then (since V is open) there is c such that a<c<b and cV. Thus UV. Contradiction. Therefore R is unboundedPlanetmathPlanetmath.

Take any unbounded sequence (an)n=1 from R. Then we have


and thus U=, so V=. This completes the proof.

Corollary. For any a,b such that a<b intervals (a,b), [a,b), (a,b] and [a,b] are connected.

Proof. One can easily show that intervals are continous image of and therefore intervals are connected.

Title intervals are connected
Canonical name IntervalsAreConnected
Date of creation 2013-03-22 18:32:49
Last modified on 2013-03-22 18:32:49
Owner joking (16130)
Last modified by joking (16130)
Numerical id 5
Author joking (16130)
Entry type Example
Classification msc 54D05