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Therefore Llx = where,/, = {(XxX)\P: P E 8 and pnLlx = 0} is a family of open subsets of X and 11'1 :s; 181 :s; T. Consequently, Ll(X) :s; 11'1 :s; T. This proves the following. Proposition 3. If X is regular then Ll(X) :s; nw(X). Theorem 1 is now a consequence of the following result (see Arhangel'skii and Ponomarev (1974». Theorem 2. For every compactum X, Ll(X) = w(X). The proof of Theorem 2 is based on the ideas of the next section. That is where we will give its outline. Corollary 1. For every compactum X, w(X) :s; IXI.

Thus the fundamental concept of the Cantor's set theory, namely the concept of a cardinal number, becomes one of main tools of general topology. It became clear that with the help of cardinal numbers it is possible to describe many phenomena that occur in topological spaces: their "size", peculiarities of their local structure, the nature of the operation of the closure and even the ability of a space to change (through the behavior of cardinal invariants under continuous mappings). For this reason cardinal invariants arise naturally and are useful in practically all areas of general topology.

Proposition 3. If f : X - ? Y is an irreducible mapping that is both continuous and closed, then the inverse image under f of a dense subset of Y is dense in X. I. Compactness 25 Theorem 15. If a compactum X can be mapped continuously and irreducibly onto a compactum Y, then the spaces X and Y have the same density and the same 7r-weight. One quite simple but fundamentally important corollary to Theorem 15 is the following: the cardinality of each compactum admitting an irreducible continuous mapping onto a compactum X is bounded by the cardinal number 221XI.

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