Definition:Higher-Aleph Complement Topology

Definition

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

Let $\tau \subseteq \powerset S$ be the set of subsets of $S$ defined as:

$\tau = \set {U \subseteq S: \size {\relcomp S U} = \aleph_m: m < n} \cup \set {U \subseteq S: \relcomp S U \text { is finite} } \cup \O$

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 $T = \struct {S, \tau}$ is an $\aleph_m$ complement space.

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

Also see

• Higher-Aleph Complement Topology is Topology

• Results about higher-aleph complement topologies can be found here.

Linguistic Note

$\aleph_n$ is read aleph n.

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

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 anyway.