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I was hoping someone could explain the following to me. My notes say: In the space of $[a,b]$, the interior of $[a,b]$ is $[a,b]$

The definition I have for interior is that the interior of a set $H$ is the set of points $x$ of which H is a neighbourhood, but yet I don't understand how this works because surely the points $a$ and $b$ can't be in a neighbourhood of H can they?

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    An entire space is always open whatever it is. – 2017-02-05
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    Of course. I completely missed that fact in the definition of a topology. Thank you – 2017-02-05

2 Answers 2

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One way to describe the interior of a set $A$ in a space $X$ is the largest open set $U$ that is a subset of $A$. In the space $[a,b]$, the open sets unions of those of the form $(x,y)\cap [a,b]$. So in particular, $[a,b]$ itself is open in $[a,b]$, as is $(x,b]$ and $[a,x)$ for any $a

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In the subspace topology, you define the open sets of the subspace to be the open sets of the greater space intersected with the subspace. For example, $(a,b)$ is open in $[a,b]$ since $$[a,b] \cap \underbrace{(a,b)}_{\text{open in} \ \mathbb{R}} = (a,b)$$

$[a,b]$ is a subspace (in a topological sense) of $\mathbb{R}$. Then you can obtain $[a,b]$ by intersecting it with any open interval in $\mathbb{R}$ greater than $[a,b]$ and this is then, by defintion, an open set in $[a,b]$.