The definition of the addition is continuous on a linear space with a topology?

Question

I saw the definition of topological vector space. Let $X$ be a linear space with a topology such that the map $X\times X\rightarrow X$, $(x,y)\mapsto x+y$ is continuous. It follows that for all $x_1$, $x_2$ $\in$ $X$, if $V$ is a neighborhood of $x_1+x_2$, then there exists a neighborhood $V_i$ of $x_i$ $(i=1,2)$ such that $V_1+V_2\subseteq V$.

How can we get this from the definition of the continuity in a topological space? Thank you in advance!

Answer 1

Let $p: X \times X \to X$ be the map $p(x_1,x_2) = x_1 + x_2$, defined on $X \times X$ with the product topology.

The latter means that a point $(x_1, x_2) \in X \times X$ has basic neighbourhoods that are of the form $V_1 \times V_2$, where $V_1, V_2 \subseteq X$ are open and $x_1 \in V_1, x_2 \in V_2$.

So the standard topological definition of continuity at $(x_1,x_2) \in X \times X$ of $p$ is translated as:

For every open neighbourhood $V$ of $x_1 + x_2 = p(x_1,x_2)$ there exists a product basic neighbourhood $V_1 \times V_2$ of $(x_1,x_2)$ such that $p[V_1 \times V_2] \subseteq V$.

It then suffices to note that $p[V_1 \times V_2] = \{p(a,b): a \in V_1, b\in V_2\} = \{a+b: a \in V_1, b\in V_2\} =V_1 + V_2$ by definition. Hence the criterion is explained.