(i) | $E \subseteq X$ is a D78: Subset |

(ii) | $\mathcal{I} : = \mathcal{I}_T(E)$ is the D3876: Set of isolated points of $E$ in $T$ |

**topologically self-dense**in $T$ if and only if \begin{equation} \mathcal{I} = \emptyset \end{equation}

▾ Set of symbols

▾ Alphabet

▾ Deduction system

▾ Theory

▾ Zermelo-Fraenkel set theory

▾ Set

▾ Subset

▾ Power set

▾ Hyperpower set sequence

▾ Hyperpower set

▾ Hypersubset

▾ Subset algebra

▾ Subset structure

▾ Topological space

▾ Isolated point

▾ Set of isolated points

▾ Alphabet

▾ Deduction system

▾ Theory

▾ Zermelo-Fraenkel set theory

▾ Set

▾ Subset

▾ Power set

▾ Hyperpower set sequence

▾ Hyperpower set

▾ Hypersubset

▾ Subset algebra

▾ Subset structure

▾ Topological space

▾ Isolated point

▾ Set of isolated points

Formulation 0

Let $T = (X, \mathcal{T})$ be a D1106: Topological space such that

Then $E$ is a **topologically self-dense** in $T$ if and only if
\begin{equation}
\mathcal{I} = \emptyset
\end{equation}

(i) | $E \subseteq X$ is a D78: Subset |

(ii) | $\mathcal{I} : = \mathcal{I}_T(E)$ is the D3876: Set of isolated points of $E$ in $T$ |