What is an open sphere? What is an open sphere? I recently came across this phrase in these notes on complex analysis (pg. 12, Lemma 2.1.3). 
I know what an open ball is, but this phrase is confusing. 
Any help would be great.
 A: Firstly we have to set in mind that mathematicians frequently use metaphors to construct their notions, which broadly means similes based on similarity (I highly recommend you to search the Aristotle 's opera Poetics and Rhetoric about metaphor).
We can ground our thought on $\Bbb R^3$ (the $3-d$ Euclidean space with the common coördinate system), where a ball is geometrically a sphere-bounded solid. Thus an open ball is the rest of a ball without the sphere (~the "spheric shell"), and as a set can be defined via $B_ε(a):= \{x\in\Bbb R^3| \|x-a\|<ε\}$, where $a$ is the centre, $ε>0$ the radius and $\|.\|$ the euclidean norm in $\Bbb R^3$. Now we can metaphorically speak about balls of $\Bbb R^n$ for all $n\in \Bbb N$ and extend the notion of ball with the well written set discription.
With the above geometric thought we derive that a sphere can be described as $S_ε(a):= \{x\in\Bbb R^n| \|x-a\|=ε\}$, namely the boundary of ball, so $S_ε(a)=\partial B_ε(a)$. But it is common in topology, real and complex analysis to use the name sphere or ball indifferently about the interior of a euclidean sphere, so -with our symbols- the open ball is the set $B_ε(a)$ and it has boundary the spheric shell $\partial B_ε(a)$ (and of course it can be arbitrarily represented as $S_ε(a), \partial S_ε(a)$).
Finally a more neutral term is the one of open neighbourhood, which can be defined as our open ball/open sphere. The notion of open neighbourhood is broader, for an author can use it to extend the notion of a open interval from $\Bbb R$ to every $\Bbb R^n$ (or even broader it can be used to denote just open sets). In this case we will get the interiors of euclidean parallelograms in $\Bbb R^2$ or parallelepipeds in $\Bbb R^3$ and so on.
A: They must mean an open ball/disc, it's the only thing that makes sense here.
I agree though that the use of the word sphere is a bit confusing, in my mind an open sphere would be the set $\{x\in\mathbb{R}^3\mid |x-a|<\varepsilon\}$ for some $a\in\mathbb{R}^3$ and $\varepsilon>0$ a real number (which makes no sense in the context of this lemma). Similarly an open disc for me is an open ball in 2 dimensions (either $\mathbb{R}^2$ or $\mathbb{C}$).
