Simplicial Structures in Topology

I.1 Topology 11
We recall that an interval of R (or more generally, of an ordered set) is a subset
A ⊂ R such that, for each a,b,x ∈ R with a < x < b, ifa,b ∈ A, wehavex ∈ A.
Real line intervals are important examples of connected spaces, as shown by the
next theorem. It is not difficult to prove that every (non-empty) interval of R must
be of one of the following types. 2 The set
[a,b]={x ∈ R | a ≤ x ≤ b}
is the closed interval with end-points a,b;theset
(a,b)={x ∈ R | a < x < b}
is the open interval with end-points a,b;thesets
(a,b]={x ∈ R | a < x ≤ b} and [a,b)={x ∈ R | a ≤ x < b}
are semi-open intervals. Thesets
(a,+∞)={x ∈ R | a < x}, [a,+∞)={x ∈ R | a ≤ x},
(−∞,b)={x ∈ R | x < b} e (−∞,b]={x ∈ R | x ≤ b}
are infinite intervals (and naturally R =(−∞,∞) is the maximal interval).
(I.1.14) Theorem. Any interval of R is a connected space.
Proof. Let X ⊂ R be a closed interval, say X =[a,b]. LetU �= /0 be a subset of X,
both open and closed; we wish to prove that U = X. SinceU is open in X, wemay
choose u ∈ U such that u ∈ ˚X. Let
s = sup{x ∈ X | [u,x) ⊂ U};
clearly, u < s. We prove that [u,s) ⊂ U. Indeed, for each v ∈ [u,s), there exists x ∈ X
such that v < x and [u,x) ⊂ U; hence, v ∈ U. We now prove that s = b. In fact, if
s �= b, thens < b; sinceU is closed in X, we conclude that s ∈ U and so, [u,s] ⊂ U.
However, U is also open and consequently there is ε > 0 such that [u,s + ε) ⊂ U;
but this contradicts the definition of least upper bound. It follows that s = b and
[u,b) ⊂ U. Similarly, (a,u] ⊂ U which implies that X =(a,b) ⊂ U ⊂ X, andwe
conclude that U = X.
The reader may verify that this proof applies to the other types of interval (open,
semi-open or infinite); alternatively, once every finite or infinite, open or closed
interval is the telescopic union of a sequence of closed intervals, for instance,
(0,1)= �
n≥2
� �
1 1
,1 − ,
n n
the remainder of the proof follows from part 2 of Theorem (I.1.13). �
2 It is enough to consider the extrema of the interval a = infA and b = sup A, if they exist. If infA
does not exist, set a = −∞;ifsupA does not exist, set b =+∞.