Continuous Lattices between T 0 Spaces 1 Grzegorz Bancerek

The terminology and notation used in this paper have been introduced in the following articles: [29], [16], [12], [13], [11], [1], [2], [32], [18], [30], [24], [25], [26], [27], [3], [9], [34], [35], [33], [28], [15], [21], [37], [10], [31], [20], [23], [5], [14], [6], [22], [8], [4], [19], [36], and [7]. Let I be a set and let J be a relational structure yielding many sorted set indexed by I. We introduce I -prodPOS J as a synonym of ∏ J. Let I be a set and let J be a relational structure yielding nonempty many sorted set indexed by I. One can check that I -prodPOS J is constituted functions. Let I be a set and let J be a topological space yielding nonempty many sorted set indexed by I. We introduce I -prodTOP J as a synonym of ∏ J. Let X, Y be non empty topological spaces. The functor [X → Y ] yields a non empty strict relational structure and is defined as follows: (Def. 1) [X → Y ] = [X → ΩY ]. Let X, Y be non empty topological spaces. Observe that [X → Y ] is reflexive transitive and constituted functions. Let X be a non empty topological space and let Y be a non empty T0 topological space. Observe that [X → Y ] is antisymmetric. We now state three propositions: (1) Let X, Y be non empty topological spaces and a be a set. Then a is an element of [X → Y ] if and only if a is a continuous map from X into ΩY.