Metric space and continuous function Background: This is an exercise problem from Munkres's Topology (Exercise 3 of Section 20 "The Metric Topology", 2nd edition). It has been posted at this site: Topology induced by metric space. However, I am confused about some basic conceptual problems which have not been mentioned there.

The Exercise: Let $X$ be a metric space with metric $d$.
  (a) Show that $d: X \times X \to \mathbb{R}$ is continuous.
  (b) Let $X'$ denote a space having the same underlying set as $X$. Show that if $d: X' \times X' \to \mathbb{R}$ is continuous, then the topology of $X'$ is finer than the topology of $X$.

I am quite confused about the underlying concepts in the exercise:    

Problem: (1) Why to use the word "if" in (b) (i.e., if $d: X' \times X' \to \mathbb{R}$ is continuous), since that any metric $d$ has been proved continuous in (1)?
  (2) In (b), are $X$ and $X'$ both metric spaces? If so, what are their metrics, respectively? If not, how to compare two topologies if one is a metric space while the other one not?


(1) The topology of $X'$ (which only shares the underlying set with $X$) is not given by the metric $d$ (which defines the topology on $X$). The exercise shos onl ythat $d$ is continuous with respect to the topology given by $d$.
(2) $X'$ is only called a "space" here. So unless this exercise is in a chapter where general topology (i.e. without metric) has not yet been introduced, I suppose tha t$X'$ is a general topological space. At any rate, the topologies of $X'$ and $X$ can be compared, both being a set of subsets of the underlying space.
