linear continuum
Let $X$ be a totallyordered set under an order $$ having at least two distinct points. Then $X$ is said to be a linear continuum if the following two conditions are satisfied:

1.
The order relation $$ is a dense total order (i.e., for every $x,y\in X$ with $$ there exists $z\in X$ such that $$).

2.
Every nonempty subset of $X$ that is bounded above has a least upper bound (i.e., $X$ has the least upper bound property).
Some examples of ordered sets that are linear continua include $\mathbb{R}$, the set $[0,1]\times [0,1]$ in the dictionary order, and the socalled long line $\mathrm{\Omega}\times [0,1)$ in the dictionary topology^{}. (The third example is a special case of a general result on wellordered sets and linear continua.)
Proposition.
If $X$ is a wellordered set, then the set $X\mathrm{\times}\mathrm{[}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{)}$ is a linear continua in the dictionary order topology.
Linear continua are of special interest when they are made into topological spaces under the order topology, and the following two establish some useful properties of such spaces:
Proposition.
As a corollary of the preceding , we obtain the result that $\mathbb{R}$ is in its usual topology, as are the intervals $[a,b]$ and $(a,b)$, where $$.
Proposition.
If $X$ is a linear continuum in the order topology, then every closed interval in $X$ is compact^{}.
Proof.
This is essentially a slightly generalized version of the HeineBorel Theorem for $\mathbb{R}$, and the proof is almost identical. ∎
Title  linear continuum 
Canonical name  LinearContinuum 
Date of creation  20130322 17:17:40 
Last modified on  20130322 17:17:40 
Owner  azdbacks4234 (14155) 
Last modified by  azdbacks4234 (14155) 
Numerical id  11 
Author  azdbacks4234 (14155) 
Entry type  Definition 
Classification  msc 06F30 
Classification  msc 54B99 
Related topic  DenseTotalOrder 
Related topic  TotalOrder 
Related topic  Supremum^{} 
Related topic  LowestUpperBound 
Related topic  OrderTopology 
Related topic  ASpaceIsConnectedUnderTheOrderedTopologyIfAndOnlyIfItIsALinearContinuum 
Defines  linear continuum 