intervals are connected
We wish to show that intervals (with standard topology) are connected. In order to this, we will prove that the space of real numbers ℝ is connected. First we need a lemma.
Let (X,d) be a metric space. Recall that for x∈X and r∈ℝ+ we have
B(x,r)={y∈X|d(x,y)<r}. |
Lemma. Let (X,d) be a metric space and R⊂ℝ+ such that R is nonempty and bounded. Then for any x∈X we have
⋃r∈RB(x,r)=B(x,sup(R)). |
Proof. Assume that y∈⋃r∈RB(x,r). Then there is r0∈R such that d(x,y)<r0 and thus d(x,y)<sup(R), so y∈B(x,sup(R)).
Now assume that y∈B(x,sup(R)). Then d(x,y)<sup(R) and it follows (from the definition of supremum) that there is r0∈R such that d(x,y)<r0 and therefore y∈B(x,r0)⊂⋃r∈RB(x,r), which completes the proof. □
Proposition. The space of real numbers is connected.
Proof. Assume that U,V⊆ℝ are open subsets of ℝ such that U∩V=∅ and U∪V=ℝ. Furthermore assume that U≠∅ and take any x0∈U. Then (since U is open) there is r0∈ℝ such that the open ball
B(x0,r0)={x∈ℝ||x-x0|<r0} |
is contained in U. Consider the set
R={r∈ℝ+|B(x0,r)⊆U}. |
Thus R is nonempty.
Assume that R is bounded. Denote by s=sup(R)<∞. We can apply the lemma:
⋃r∈RB(x0,r)=B(x0,s). |
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
B(x0,s)=(a,b) |
for some a,b∈ℝ. Since (a,b) is maximal then a∉U or b∉U. Indeed, if both a∈U and b∈U, 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. Contradiction.
Without loss of generality we can assume that b∉U. Then b∈V, because U∪V=ℝ. But then (since V is open) there is c∈ℝ such that a<c<b and c∈V. Thus U∩V≠∅. Contradiction. Therefore R is unbounded.
Take any unbounded sequence (an)∞n=1 from R. Then we have
ℝ=∞⋃n=1B(x0,an)⊆U |
and thus U=ℝ, so V=∅. This completes the proof. □
Corollary. For any a,b∈ℝ 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 ℝ and therefore intervals are connected.
Title | intervals are connected |
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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 |