# Connected subsets of lines

For me every connected set has at least two elements.

Connected subsets of the real line have non-empty interiors. I am curious if the same is true for connected subsets of any connected linearly ordered topological space (the topology comes from the order).

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Yes. In fact, the space need not be connected, as all connected subsets of a linearly ordered topological space are intervals, and in a linearly ordered space connected intervals have nonempty interior. Let $X$ be such a space. To prove the first statement, assume $S$ is connected but not an interval, so we have some $a,b\in S$ and $x\in X$ such that $a<x<b$ but $x\notin S$. Then the sets $(-\infty, x)\cap S$ and $(x,\infty)\cap S$ are open in $S$, nonempty, and their union covers $S$, contradicting the fact that $S$ is connected. To prove the second, let $I$ be a connected interval in $X$ with at least two points $a,b\in I$, chosen so that $a<b$. Then the open set $(a,b)$ is contained in $I$, and the set $(a,b)$ cannot be empty, as otherwise we would have that $$I = \big((-\infty,b)\cap I\big)\cup\big((a,\infty)\cap I\big)$$ is disconnected as these are disjoint nonempty relatively open sets. Thus $I$ has nonempty interior.
The argument isn’t quite complete: $(a,b)$ could be empty. But in that case $\{a\}$ and $\{b\}$ separate $[a,b]$. –  Brian M. Scott Dec 26 '11 at 19:41
Thank you. By the way, assuming the Souslin line exists, must its density character be $\omega_1$? –  S. Serva Dec 26 '11 at 19:59
@S.Serva: Yes. Every Suslin line contains an Aronszajn line as a dense subset, and every Aronszajn line has cardinality $\omega_1$. –  Brian M. Scott Dec 26 '11 at 20:17