EDB — 0HW

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

Let \((I,≤)\) be a set with direct ordering and with a maximum that we call \(∞\). We call \(J=I⧵\{ ∞\} \) 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:

• \(∅,I\);

• sets \(A\) that contain a ”half-line” \(\{ k∈ I : k≥ j\} \), for a \(j{\lt}∞\), (these are called “neighborhoods of \(∞\)”);

• subsets of \(I\) that do not contain \(∞\).

Show that \(𝜏\) is a topology. Is this topology Hausdorff? Show that \(∞\) is the only accumulation point.