Fractal geometry: Is A+1 compact if A is a nonempty compact set?

Question

Define: Let $(X,d)$ be a metric space and $\mathscr H$ be the collection subset of X other than the empty set.

Define: Let $(X,d)$ be a matric space and $S\subset C$. For $\epsilon\ge0$, define $$S+\epsilon:=\{y\in X\;|\; d(x,y)\le\epsilon\ for\ some\ x\in S \}$$

Here comes my question,

Let $(X,d)$ be a metric space and $A\in\mathscr H(X)$. Is $A+1$ always compact?

I think it's yes, because I think $A+1$ has no big difference with $A$. But I don't know how to prove it.

Am I wrong? please help me, thanks.

Answer 1

Let $(X,d)$ be the space $[0,1[$ with the usual distance. Whatever may be your definition of $\mathscr H$ (non empty compact sets or just non empty sets) the set $A = \{0\}$ is compact, and $A + 1 = X$ is not compact.