linear continuum

Let $X$ be a totally-ordered 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 $x<y$ there exists $z\in X$ such that $x<z<y$ ).
2.

Every non-empty 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 so-called long line $\Omega\times[0,1)$ in the dictionary topology. (The third example is a special case of a general result on well-ordered sets and linear continua.)

Proposition.

If $X$ is a well-ordered set, then the set $X\times[0,1)$ 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.

If $X$ is a linear continuum in the order topology, then $X$ is http://planetmath.org/node/941connected and so are intervals in $X$ .

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 $a<b\in\mathbb{R}$ .

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 Heine-Borel Theorem for $\mathbb{R}$ , and the proof is almost identical. ∎

Title	linear continuum
Canonical name	LinearContinuum
Date of creation	2013-03-22 17:17:40
Last modified on	2013-03-22 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