EDB — 0HW

[0HW] Topics:direct ordering. Prerequisites:[06M]↺↻, [06N]↺↻, [06V]↺↻.

Let $(I,\le )$ be a set with direct ordering and with a maximum that we call $\infty$. We call $J=I⧵\{\infty \}$ and assume that $J$ is filtering (with induced sorting) and non-empty. In this case we propose a finer topology. The topology $𝜏$ for $I$ contains:

• $\varnothing ,I$;

• sets $A$ that contain a ”half-line” $\{k\in I:k\ge j\}$, for a $j<\infty$, (these are called “neighborhoods of $\infty$”);

• subsets of $I$ that do not contain $\infty$.

Show that $𝜏$ is a topology. Is this topology Hausdorff? Show that $\infty$ is the only accumulation point.