Exercise 9, § 3

Let $A$ be a subset of a linearly ordered set $X$.

a) Show that the topology induced on $A$ by the topology $\mathfrak{T}_-(X)$ [resp. $\mathfrak{T}_+(X), \mathfrak{T}_0(X)$] (§ 2, Exercise 5) is finer than the topology $\mathfrak{T}_-(A)$ [resp. $\mathfrak{T}_+(A), \mathfrak{T}_0(A)$], the ordering of $A$ being that induced by the ordering of $X$. If $A$ is an interval of $X$, then $\mathfrak{T}_-(A)$ [resp. $\mathfrak{T}_+(A), \mathfrak{T}_0(A)$] is the topology induced on $A$ by $\mathfrak{T}_-(X)$ [resp. $\mathfrak{T}_+(X), \mathfrak{T}_0(X)$].

b) Take $X$ to be the product $\mathbf{Q} \times \mathbf{Z}$, ordered by the lexicographic ordering (Set Theory, Chapter III, § 2, no. 6) and take $A$ to be the subset $\mathbf{Q} \times \{0\}$. Show that the topology $\mathfrak{T}_-(A)$ [resp. $\mathfrak{T}_+(A), \mathfrak{T}_0(A)$] is strictly coarser than the topology induced on $A$ by $\mathfrak{T}_-(X)$ [resp. $\mathfrak{T}_+(X), \mathfrak{T}_0(X)$].