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24.  Connected Subspaces of the Real Line.

Defn:  A simply ordered set $L$ having more than one element is
called a {\bf linear continuum} if the following hold:
\v
(1) $L$ has the least upper bound property.
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(2)  If $x < y$, there exists $z$ such that $x < z < y$


Thm 24.1:  If $L$ is a linear continuum in the order topology, then $L$ is connected, and so are intervals and rays in $L$.

Cor 24.2:  The real line is connected and so are intervals and rays in ${\cal R}$.

Thm 24.3 (Intermediate value theorem).  Let $f: X \rightarrow Y$ be a continuous map where $X$ is connected and $Y$ is an ordered set with the order topology.  If $a, b \in X$ and if $r \in Y$ is a point lying between $f(a)$ and $f(b)$, then there exists a point $c$ of $X$ such that $f(c) = r$.

Defn:  Given points $x, y \in X$, a {\bf path} in $X$ from $x$ to $y$ is a continuous map $f:[a, b] \rightarrow X$ such that $f(a) = x$ and $f(b) = y$.

A space is {\bf path connected} if every pair of points of $X$ can be joined by a path in $X$.

Lemma:  A path connected space is connected.

Lemma:  There exists a connected space which is not path connected.

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