Definition:Higher-Aleph Complement Topology

Definition
Let $S$ be a set whose cardinality is $\aleph_n$ where $n > 0$.

Let $\tau \subseteq \mathcal P \left({S}\right)$ be the set of subsets of $S$ defined as:
 * $\tau = \left\{{U \subseteq S: \left\vert{\complement_S \left ({U}\right)}\right\vert = \aleph_m: m < n}\right\} \cup \left\{{U \subseteq S: \complement_S \left ({U}\right) \text { is finite}}\right\} \cup \varnothing$

That is, $\tau$ is the set of subsets of $S$ whose complements relative to $S$ are of a cardinality strictly less than $S$.

Then $\tau$ is an $\aleph_m$ complement topology on $S$, and the topological space $T = \left({S, \tau}\right)$ is an $\aleph_m$ complement space.

This construction is an extension of the concept of the finite complement topology and the countable complement topology.

Linguistic Note
$\aleph_n$ is read aleph n.

The symbol $\aleph$ (aleph) is the first letter of the Hebrew alphabet.

Also see

 * Higher-Aleph Complement Topology is a Topology

Note
The author of this page has never seen this concept in anything he has read. This may be because of:
 * a) his limited reading materials, in which case he needs to be enlightened,
 * b) because this concept has genuinely not been thought of before, in which case he claims precedence, or
 * c) because the concept isn't actually worth writing down as it doesn't lead anywhere, in which case the author reserves the right to explore it himself.