$f,g: X \to A$ be continuous functions then $f+g$ is also continuous.

Question

Let $X$ be a topological space and $A$ be an abelian group with discrete topology. If $f,g: X \to A$ are continuous functions then how do I show that $f+g$ is also continuous ?

So I need to show that $(f+g)^{-1}\{a\}$ is an open subset of $X$ for all $a \in A$. I spent a lot of time but no result. please help.

Answer 1

An alternative approach (and arguably easier) is to utilize compositions of continuous functions.

So first you show that

$$f\times g:X\to A\times A$$ $$f\times g(x)=\big(f(x), g(x)\big)$$

is continuous. This follows because $\pi_1\circ(f\times g)=f$ and $\pi_2\circ(f\times g)=g$ are continuous for both projections $\pi_i$. This is the universal property of products.

Then your $f+g$ is the composition $+\circ (f\times g)$, where $+:A\times A\to A$ is addition, which is continuous because the topology on $A$ is discrete.

Note that this generalizes to any topology on $A$ as long as $+:A\times A\to A$ is continuous.