# 3-D Fracture Propagation Simulation and Production by Guo D., Ji L. By Guo D., Ji L.

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We claim that f −1 (O) ∈ U for any open set O intersecting the set F (U). / U and using the unsplit property of U we shall get U = Assuming that f −1 (O) ∈ ω \ f −1 (O) ∈ U and thus f (U ) ⊂ X \ O. Then clX (f (U )) ⊂ clX (X \ O) = X \ O. Consequently, F (U) ⊂ clX (f (U )) ⊂ X \ O does not intersect the set O, which is a contradiction. Next, we show that the set F (U) is a singleton. Assuming the converse we can ﬁnd two open disjoint sets O1 , O2 intersecting F (U). Then their preimages f −1 (O1 ) and f −1 (O2 ) are disjoint are belong to U, which is a contradiction.

Next, we show that πχ(U) ≥ cf(c). Assuming that πχ(U) < cf(c), ﬁx a family B ⊂ [ω]ω of size |B| < cf(c) such that each U ∈ U contains some B ∈ B. Since B ⊂ [ω]ω = {Bα : α < c} and |B| < cf(c), there is an ordinal β < c such that B ⊂ {Bα : α < β}. By the construction of the function f , the set Aβf(β) ∈ X ⊂ U contains no set Bα with α < β and thus contains no B ∈ B, which contradicts the choice of B. Next, we introduce two cardinal characteristics allowing us to measure how non-linked a semiﬁlter is.

The norm R of a relation R ⊂ X × Y is the smallest size |D| of an R-dominating subset D in Y . If X = Y we shall write (X, R) instead R to avoid possible misunderstanding. 6. Verify that d = (ω ω , ≤∗ ) , b = (ω ω , ≥∗ ) , s = ([w]ω , ) , r = ([ω]ω , ) . 7. Often dominating sets in posets are referred to as coﬁnal sets. The smallest size of a coﬁnal subset in a poset (X, ≤) is called the coﬁnality of (X, ≤). Observe that the coﬁnality of (X, ≤) is nothing else but the norm (X, ≤) . There are also small cardinals that cannot be deﬁned with help of norms, for example so are the small cardinals p, t, h, and g.

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