Why are the interior points in this question not the same? I'm working on a question that wants me to write down the interior points of an interval contained in a metric space.
$Let X=((1,7],d_{E})$ be a subspace of the metric space $(\mathbb{R},d_{E})$.
Let $A=[5,7]$. Find the interior points of A regarded as a subset of


*

*$X$

*$(\mathbb{R},d_{e})$


My answer is: (5,7) for both. The question however also states that they are not the same. How so? What am i misunderstanding?
Regards
 A: This is because, in the first case, the interior points are $(5,7]$. 
Indeed, $(5,7]=(5,8)\cap (1,7]$, so it is open in $X$.
And clearly $5$ is not in the interior of $[5,7]$, since every open neighborhood contains points smaller that $5$.
Note: see here for the notion of subspace topology, to see in particular why I claim that $(5,7]$ is open in $X$: http://en.wikipedia.org/wiki/Subspace_topology
Alternative: write
$$
(5,7]=\{x\in X\;|\; d_E(x,6.5)<1.5\}
$$
to see that this is open in $X$, as an open ball.
A: The interior of a set, $\operatorname{Int}(A)$ is usually defined to be the union of all open subsets contained in the set $A$. 
In this case, if you work with the induced subspace topology on $X$ then, for example, $(6,8)$ is open in $\mathbb{R}$. Hence $(6,8)\cap X = (6,7]$ is open in $X$. Hence $7$ is in the interior of $(5,7]$. 
Even if you prefer working with the just $(X,d_E)$ as a metric space in it's own right and not as a subspace, then it's still true that $(6,7]$ is open in $X$ since it is the open ball around $7$ of radius 1.
I hope that's clear now?
