# Definition:Irreducible Subset (Topology)

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## Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $A$ be a subset of $S$.

Then $A$ is **irreducible (subset)** if and only if

- $A$ is non-empty and closed and for all closed subsets $B, C$ of $S$: $A = B \cup C \implies A = B$ or $A = C$

## Sources

- Mizar article YELLOW_8:def 3