What is the direct sum of a singleton and an interval?

Question

Let $\{a \}\subseteq \mathbb{R}$, and $A \subseteq \mathbb{R}$

What does it mean for $\{a\} \oplus A$?

Answer 1

Vectors of the form $a + a'$ for $a' \in A$ where $a$ not in $ A$.

Answer 2

As the question is tagged general-topology, I assume we are to take the direct sum in the category Top of topological spaces and continuous maps. In that case, "direct sum" is simply "disjoint union". So if $a$ is not in the closure of $A$, we can simply take $X=A\cup\{a\}$; otherwise or just to be on the safe side, take $X=A\cup\{\alpha\}$, where $\alpha$ is a symbol that is not even $\in\Bbb R$ and declare $U\subseteq X$ open iff $U\cap A$ is open.