# 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