= Coordinate seminorms
{title2=$p_x(x,y)=|x|,\quad p_y(x,y)=|y|$}
On $\mathbb R^2$, the <coordinate seminorms> $p_x$ and $p_y$ measure horizontal and vertical components separately. Their <kernels of seminorms> are the vertical and horizontal axes. Neither is a <norm>, but together they form a <separating family of seminorms>. A basic zero <neighbourhood> is
$$
\{p_x<a\}\cap\{p_y<b\}=(-a,a)\times(-b,b)\qquad(a,b>0).
$$
Each condition alone is an unbounded strip; their intersection is an open rectangle. For $a=b=1$, this is the unit <open ball> of the <maximum norm> $\|(x,y)\|_\infty=\max\{|x|,|y|\}$. For unequal tolerances it is the unit <open ball> of the weighted <norm> $\max\{|x|/a,|y|/b\}$. Thus the two coordinate measurements generate the usual <topology> of the plane.
For a numerical check, take $a=1$, $b=1/2$: $(0.8,0.4)$ passes both tests, $(0.8,0.7)$ passes only the horizontal test, and $(1.2,0.4)$ passes only the vertical test. A direction missed by one <seminorm> is detected by the other.
\Image[/seminorm-neighborhoods.png]
{title=Two coordinate seminorm strips intersect in a basic open rectangle}
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