= Euclidean topology
{c}
{title2=$\tau_{\|\cdot\|_2}$}
The Euclidean topology on $\mathbb R^n$ is the <metric topology> induced by the <Euclidean norm>. A <subset> is <open> exactly when each of its points has an <open ball> contained in it. It agrees with the finite <product topology> on $n$ copies of the <real line>: the inequalities $\|x\|_\infty\leq\|x\|_2\leq\sqrt n\|x\|_\infty$ put a coordinate box inside every sufficiently small ball and a ball inside every sufficiently small coordinate box. The case $n=0$ is the one-point <topological space>.
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