Continuity of addition under the lower limit topology

Question

Letting $\Bbb R_\ell$ denote $\mathbb{R}$ with the lower limit topology, prove that the function $f : \Bbb R_\ell \times \Bbb R_\ell \to \Bbb R_\ell$ defined by $f((x,y)) = x+y$ is continuous.

Answer 1

The preimage of an open set $[a,b)$ is $$ \{(x,y)\colon a \le x+y < b\}. $$ To prove that such a stripe is open in the product space, write it as the union of squares $[x,x+(b-a)/2) \times [a-x,a-x+(b-a)/2)$ for $x \in \mathbb R$.

Answer 2

Computational error corrected.

Start with a basic open set in the range, say $[a,b)$. Its inverse image under addition is

$$\{\langle x,y\rangle\in\Bbb R^2:a\le x+y<b\}\;;$$

geometrically this is the diagonal strip between the lines $x+y=a$ and $x+y=b$, including the former edge but not the latter. Take a point $p=\langle x_0,y_0\rangle$ on the line $x+y=a$. The distance from $p$ to the line $x+y=b$ is $b-a$, but that’s in the direction perpendicular to the strip, i.e., in the direction parallel to the line $y=x$. Thus, the point on the line $x+y=b$ closest to $p$ is the point

$$q=\left\langle x_0+\frac1{\sqrt2}(b-a),y_0+\frac1{\sqrt 2}(b-a)\right\rangle\;,$$

and $p$ and $q$ are diagonally opposite corners of the open box

$$\left[x_0,x_0+\frac1{\sqrt2}(b-a)\right)\times\left[y_0,y_0+\frac1{\sqrt2}(b-a)\right)$$

in the Sorgenfrey topology. Now just check that the strip is the union of all such open boxes.