Exercise 5, § 2

On an ordered set $X$ let $\mathfrak{T}_0(X)$ (resp. $\mathfrak{T}_+(X)$, $\mathfrak{T}_-(X)$) denote the topology generated by the set of all open intervals (resp. the set of right half-open intervals, the set of left half-open intervals), bounded or not.

a) Show that if $X$ is linearly ordered, then the open (resp. right half-open, left half-open) intervals form a base of $\mathfrak{T}_0(X)$ (resp. $\mathfrak{T}_+(X)$, $\mathfrak{T}_-(X)$); and that the closed intervals are closed sets in each of these three topologies.

b) On a linearly ordered set $X$, show that the topology $\mathfrak{T}_+(X)$ is finer than the right topology ($§ 1$, Exercise 2), that the topology $\mathfrak{T}_-(X)$ is finer than the left topology, and that the topology $\mathfrak{T}_0(X)$ is the intersection of $\mathfrak{T}_+(X)$ and $\mathfrak{T}_-(X)$.

c) Let $X$ be a linearly ordered set. Show that a necessary and sufficient condition for $\mathfrak{T}_0(X)$ to have a countable base is that there should be is a countable subset $D$ of $X$ such that, for each $x \in X$ and each interval $]a, b[$ containing $x$, there exist $\alpha \in D$ and $\beta \in D$ such that

[cf. Chapter IV, $§ 2$, Exercise 11 c)].

d) Show that if $X$ is well-ordered (Set Theory, Chapter III, $§ 2$) then the topology $\mathfrak{T}_+(X)$ is the discrete topology, and the topologies $\mathfrak{T}_0(X)$ and $\mathfrak{T}_-(X)$ are identical.