# Should the set of Continuous Functions on an Open Interval be a Vector Space?

Just a quick question. Should the set of continuous functions on an open interval be a vector space?
In all my lecture notes, such like vector spaces and real analysis. They all claim that $$\mathcal C[a,b]:=\{f:[a,b]\rightarrow \mathbb{K}|\,f\text{ is continuous}\}$$ to be a vector space. I was thinking that what if we change the closed interval to open interval, will the space still be a vector space? Since $\textit{f}$ is defined only on $(a,b)$, we just have to restrict ourselves on the open interval without discussing the $f$'s behavior on outer interval. Then it follows every condition to form a vector space.
Why cannot $\mathcal{C}(a,b)$ be an example for a vector space consists of continuous function?

## 1 Answer

For any topological space $X$, the set of continuous maps $X\to \Bbb K$ is a $\Bbb K$-vector space. This just says that the sum of continuous functions is continuous, and that multiplication with a constant is a continuous map $\Bbb K\to\Bbb K$.