Convex set definition Munkres defines thus: Given an ordered set $X$ a subset $Y$ of $X$ is convex in $X$ if for each pair of points $a&lt;b$ of $Y$ the entire interval $(a,b)$ of points of $X$ lies in $Y$. But again in Connected subspaces of real line section he recalls the definition of convex but now using a closed interval $[a,b]$. Is it $(a,b)$ in the definition or $[a,b]$?
As noted in the comments, when $a$ and $b$ are distinct points in $Y$ so knowing that $a,b\in Y$ tells you that both $(a,b)$ and $[a,b]$ lie in $Y$ and so there's no particular need to distinguish between the two.
There is one subtlety though: if $a=b$ in $Y$ then $(a,b)=\emptyset$ and $[a,b]=\{a\}$ (or $\{b\}$ if you prefer) and then it may make a difference whether you talk about open or closed intervals.  In this case $a$ is referred to as an extreme point of $Y$ and it becomes interesting to consider if a set if the closed convex hull of all of its extreme points (for example).
Speaking of examples:  the unit ball in $\ell_2$ is convex and possesses no extreme points, whereas the unit ball in $\ell_1$ is convex and does possess extreme points as you should be able to easily verify.
As noted in the comments, when $a$ and $b$ are distinct points in $Y$ so knowing that $a,b\in Y$ tells you that both $(a,b)$ and $[a,b]$ lie in $Y$ and so there's no particular need to distinguish between the two.
There is one subtlety though: if $a=b$ in $Y$ then $(a,b)=\emptyset$ and $[a,b]=\{a\}$ (or $\{b\}$ if you prefer) and then it may make a difference whether you talk about open or closed intervals.  In this case $a$ is referred to as an extreme point of $Y$ and it becomes interesting to consider if a set if the closed convex hull of all of its extreme points (for example).
Speaking of examples:  the unit ball in $\ell_2$ is convex and possesses no extreme points, whereas the unit ball in $\ell_1$ is convex and does possess extreme points as you should be able to easily verify.