## Lattice Characterizations of Topologies and Compactifications |

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a a i ak+i atom c a d c v d chapter cl(f cl(p cl(q cl(x cl(y closed sets closed subsets compact element compact neighbourhood compact set compact space complete distributive lattice consider consisting containing continuous function Conversely define definition denotes the complement disjoint equivalent family F finite intersection property following properties form a base function g Hausdorff compactification hence cl homeo homeomorphic with TL(X implies lattice of closed lattice product lattice sum mapping maximal ideal morphic morphisms natural transformation non-empty nonzero irreducible elements point closure prime ideals proefschrift proper superdense subspaces properties Bl Proposition prove quotient space Remark Similarly space satisfies strongly irreducible elements sublattice of L(X subspace of TL(X suppose surjective T-lattice T0 space theorem Top0 topological space topology zero and unit zero filter zeroset