Name of the union of a set with its holes Given an arbitrary connected and compact set $S$ with holes in it, is there a name for the simply connected set formed by the union of $S$ and its holes?

For example, let $S = \{x\in \mathbb{R}^n\ |\ 0 &lt; a \leq||x||^2_2 \leq b\}$, its hole is $H = \{x\in \mathbb{R}^n\ |\ ||x||^2_2 &lt; a\}$. Is there a name for $S \cup H$?

If there is not a specific definition for it, can I just call this set "the union of $S$ and its holes", or is there a more precise way of describing it?

By a set with holes I refer to sets such as this, which is taken from the Wikipedia page on simply connected sets. Here is my attempt to properly define a hole (at least in $\mathbb{R}^n$), inspired by this answer to another question.

Given a connected and compact subset $S \subset \mathbb{R}^n$, its boundary $\partial S$ can be written as the finite union of connected and compact hypersurfaces $F_i \subset \mathbb{R}^{n-1}$, such that $\partial S = \bigcup_{i=1}^k F_i$. Each $F_i$ cuts $\mathbb{R}^n$ in exactly two pieces $A_i \subset \mathbb{R}^n$ and $B_i \subset \mathbb{R}^n$, such that the closure of $A_i$ is compact and the closure of $B_i$ is not compact. Then, we say that $A_i$ is inside $F_i$, and $B_i$ is outside. A hole $H_i$ of $S$ is any $A_i$ such that $A_i \cap S = \emptyset$. Then, the set of all holes is $H = \bigcup H_i$.

If $S \cup H = S$, then is said that $S$ has no holes. This condition is equivalent to $H=\emptyset$.

If $S \cup H \neq S$, then I want to know if $S \cup H$ has a proper name. This condition is equivalent to $H\neq\emptyset$, and $H$ is "filling the holes" of $S$.
I assume you mean a compact set in $\mathbb{R}^n$ (or some metric space) so that "bounded" makes sense. Also, that in your clarification you probably meant to say something like the following (but see the NOTE at the end)


  "take a COMPACT simply connected set $T$ and remove ( open) subsets of the interior to get a compact connected subset $S$."


The goal being to describe $T$ in relation to $S.$ 

sandwich gives a good answer and from it we see that the key is that "hole" is something like "(a subsequently removed) bounded component of the interior of $T$ " Which becomes "bounded component of the  complement of $S.$"

One could avoid double complements by saying "the union of $S$ and the bounded components of the complement of $S.$" That makes sense even in (some) situations which are not compact. For example 


  "Let $T$ be a (finite union of) CLOSED simply connected set(s) and remove BOUNDED subsets of the interior.." 


Of course that is the same as "the complement of (the union of) the unbounded component(s) of the complement of $S.$"

NOTE: Really I (and you) should say something like "remove a finite number of open subsets each of which has compact closure."  Although one could also allow the case of an infinite number of holes, perhaps with disjoint closures. It depends somewhat on the exact situation you wish to model. 

LATER: In answer to your actual question: I am not aware of a name for what you want, although that does not at all imply there isn't one. You could call it the simply connected closure and define it as the intersection of all simply connected closed sets containing $S.$ That would definitely work provided that such an intersection is closed and simply connected. It does not immediately give one an intuitive picture of what that thing is.
A concise definition of the object you describe is "the complement of the unbounded component of the complement of $S$".  

In two dimensions, the term simply connected hull is sometimes used: for example, in a MathOverflow post and in other places you can find with a search.  

But in dimensions $\ge 3$ "simply connected" would be misleading: filling in a torus $\mathbb{T}^2$ creates a solid torus, which is still not simply connected.

Another term in use is filled-in set, for example Filled Julia set. This one can be used in any dimension.