Any nonempty open subset of an irreducible topological space is dense and irreducible. If \(Y\) is a subset of a topological space \(X\) which is irreducible in its induced topology, then the closure \(\bar{Y}\) is also irreducible.
Irreducible is strictly stronger than connected: connected forbids a decomposition into two proper disjoint closed sets, irreducible forbids it without disjointness. \(V(xy)\) is connected and reducible.
Solution
Density. Let \(U \subseteq X\) be open and nonempty with \(X\) irreducible, and suppose \(\cl_X(U) \neq X\). Then \[ X = (X \sm U) \union \cl_X(U) \] writes \(X\) as a union of two proper closed subsets, contradicting irreducibility.
Closures. Suppose \(Y\) is irreducible and write \(\cl_X(Y) = A \union B\) with \(A, B\) closed in \(X\). Intersecting with \(Y\) gives \(Y = (A \intersect Y) \union (B \intersect Y)\), so by irreducibility \(Y \subseteq A \intersect Y \subseteq A\) after relabelling. Then \(\cl_X(Y) \subseteq \cl_X(A) = A\), so the decomposition was trivial and \(\cl_X(Y)\) is irreducible.
Source Collections
Dependencies
D-9DIKBIrreducible and Noetherian spacesuses
Backlinks
P-AGXMISCIRROPENCONNA space is irreducible exactly when its nonempty open subsets are connectedrelated-to