Definition:Fortissimo Space
Definition
Let $S$ be an uncountably infinite set.
Let $p \in S$ be a particular point of $S$.
Let $\tau_p \subseteq \powerset S$ be a subset of the power set of $S$ defined as:
- $\tau_p = \leftset {U \subseteq S: p \in \relcomp S U}$ or $\set {U \subseteq S: \relcomp S U}$ is countable (either finitely or infinitely)$\rightset {}$
That is, $\tau_p$ is the set of all subsets of $S$ whose complement in $S$ either contains $p$ or is countable.
Then $\tau_p$ is a Fortissimo topology on $S$, and the topological space $T = \struct {S, \tau_p}$ is a Fortissimo space.
Also see
- Results about Fortissimo spaces can be found here.
Source of Name
This entry was named for Marion Kirkland Fort, Jr.
Linguistic Note
The name Fortissimo space is a pun on Fort space.
The musical term forte means to be played loudly.
The term fortissimo is a musical term meaning to be played very loudly.
So in a sense the term Fortissimo space can be thought of as being:
- like a Fort space, but more so.
As the term ultimately derives from a proper noun, that is the surname of Marion Kirkland Fort, Jr, the word Fortissimo when used in this context is capitalised.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $25$. Fortissimo Space