APPENDIX I PARACOMPACT SPACES In this appendix we discuss some of the properties of paracompact spaces, originally defined by Dieudonne [4J. Definition 1. Let 'U = {Ua I a E A} be an open covering of a topological space X. A partition of unity subordinated to 'U is a collection {Pa I a E A } of continuous functions pa: X ~I such that (i) for each point x of X, there is a neighborhood on which all but a finite number of the pa vanish; (ii) La pa = 1 at each point x of X; (iii) for each aEA, the function pa vanishes outside Ua. A covering 'U = {U a I a E A} of X is numerable if there is a sub­ ordinated partition of unity p = {Pa I a E A}. The functions p deter­ mine a map p:X ~ .6.('U) into the simplex of 'U, in fact, into the sub­ complex N('U), the nerve of 'U (see IV.4). As observed in IV.4, the fact that p is a locally finite family of functions implies that each x E X has a neighborhood Vx such that p(Vx) lies in a finite subcomplex of .6.('U). For Vx we choose a neighborhood of x on which only finitely many functions pa, say pall···' pan! are nonzero, so that p(Vx) lies in the simplex with vertices corresponding to Uau ••• , Uan• Thus p(Vx ) has a compact closure in .6.('U) , and p I Vx is continuous when .6.('U) is given any topology, such as the OW or metric topology, which re­ duces to the simplicial topology on its finite subcomplexes. Definition 2. A map p:X ~ Y is compact if the image of p has compact closure in Y and is locally compact if each point of X has a neighborhood the image of which has compact closure in Y. A map p:X ~ .6.('U) into a simplex is full if the image of p is not contained in any subsimplex. 201 202 PARACOMPACT SPACES App.I From the discussion above we have the following. PROPOSITION 3. The partition8 of unity on X corre8pond biuniquely to the full locally compact map8 of X into CW simplexe8 A (a) . I A Hausdorff space X is paracompact if each open covering has a sub­ ordinated partition of unity. We note that this is not the usual definition of paracompactness, but is equivalent to it (see proposition 4 below). Also, this definition differs from the one given in Kelley [22J in that it requires X to be Hausdorff. Example8. 1. Any compact Hausdorff space. 2. Any OW complex (see 11.4 and Miyazaki [31J). 3. Any metric space (see Kelley [22, p. 160J). 4. Any Hausdorff Lindelof space (see Dugundji [8, p. 174J). This includes the fact that a Hausdorff u-compact space is paracompact. We observe that a paracompact space is normal. Given two closed disjoint sets C and D in X, the complements form an open cover of X. In a subordinated partition of unity {Pl,P2}, one function must be 0 on D and 1 on C, and thus must be a Urysohn function for the pair. Since we assume a paracompact space is Hausdorff, it must therefore be normal. Given a paracompact space X and an open covering 'U = { Ua}, there are several special types of refinements which we can obtain. These can be constructed easily given a partition of unity subordinated to 'U. PROPOSITION 4. Let X be a topological 8pace, let 'U = { U a} be an open covering of X, and let {Pal be a partition of unity 8ubordinated to 'U. Then (i) 'U has a locally finite open refinement; (ii) 'U has a locally finite cl08ed refinement; (iii) 'U has an open 8tar refinement. In each ca8e, the8e refinement8 have 8ubordinated partition8 of unity. Proof. The family {Pal of continuous functions determine a map p: X --+ A ('U) into the simplex of 'U. If Ua denotes the vertex correspond- App. I PARACOMPACT SPACES 203 ing to Ua, then pa (x) represents the barycentric coordinate corresponding to Ua • Thus when the simplex is given the metric topology, the map p is continuous, because the barycentric coordinate functions compose with the map p to give continuous functions. The pre-image under p of st Ua is an open subset of Ua • The pre-image under p of a locally finite open or closed refinement or of a star refinement is a refinement of the same nature of {U a}. Hence we have only to construct a refinement of the type desired for the covering {st Ua } of A('U), and p pulls it back to an appropriate refinement of 'U. If we construct such a refinement with a subordinated partition of unity, this will induce one on X with a subordinated partition of unity. (i) The cover Va={xlba(x»!SUPB{b~(x)}} is a locally finite open refinement of {st Ua }, where ba is the barycentric coordinate function corresponding to Ua • The subordinated partition of unity is given by: The corresponding locally finite open refinement of 'U is {U:}, where U:={x I Pa(X»! sup~ {p~(x)}}, and the subordinated partition of unity is {PaP}. (ii) Take the closures of the sets Va. Then Va= {x I ba(x) ~ ! sup~ {b~(x)}} is a locally finite closed refinement of {st Ua}. (iii) Take the barycentric subdivision of A('U) and apply (i) to the covering by stars. I LEMMA 5. (shrinking lemma). In a normal space X, any locally finite open covering 'U has a closed locally finite refinement. Proof. Well order the sets Ua of the covering 'U. If we have found open sets Va (a<f3) such that VaCUa for each a<f3 and {Va I a<f3}U {Ua I a~f3} is an open covering of X, then let U; = U~-U {Va I a<f3}­ U{Ua I a>f3}. Since U~ is a neighborhood of U; in X, it follows that U~~ 0;. Choose an open set V~~ 0; whose closure lies in U~. Then {Va I a::::;f3}U{Ua I a>f3} is an open covering of X and VaCU" for a::::;f3. The lemma then follows by transfinite induction. Note that since Ua~V" and {Ua} is locally finite, then {Va} and {Va} are also locally finite. I 204 P ARACOMPACT SPACES App. I PROPOSITION 6. A Hausdorff space X is paracompact if and only if each open cover has a locally finite open refinement. Proof. The implication Honly if" is proved above. To prove the converse, first we prove that a Hausdorff space with the indicated property is normal. This is done by an obvious modification of the proof that a compact (Hausdorff) space is normal, keeping in mind the fact that the union of closures of a locally finite collection of sets is the closure of the union. Next apply the shrinking lemma to any open locally finite refinement of any prescribed open cover {U a} of X. Let {Va} be the resulting closed refinement. For each a let U a be an Urysohn function 0 off U a and 1 on Va. The functions {ua} are a family of nonnegative functions with locally finite supports, hence the sum Q= L U a is every­ where continuous and bounded; since every point lies in some V", the sum is positive as well at every point. Therefore {u,,/Q} is a family of continuous functions on X. It is clearly a partition of unity subordinated to {U,,}. I Definition 7. A Hausdorff space X is perfectly normal if for each closed subset C, there is a continuous nonnegative real valued function on X for which C is the set of its zeros. This is a strengthening of the condition of normality; for if C and C' are disjoint closed sets, in a perfectly normal space let f,y vanish on C,C' and nowhere else. Then U= f(1-y)/sup (f,y) is a Urysohn function for C and C'. Not every normal space is perfectly normal. In Dugundji [8, p. 148J it is proved that a closed set C in a normal space is the set of zeros of a continuous real valued function if and only if it is a Go, that is, the intersection of a decreasing family of countably many open sets each containing the closure of the following. Hence in a normal space that does not satisfy the first axiom of countability, some point is a closed set which is not a Ga. It follows from this criterion that any metric space is perfectly normal. Unlike normality, perfect normality is clearly hereditary; any subspace of a perfectly normal space is again perfectly normal. The following lemmas lead to a proof of theorem 10. LEMMA 8. Every subspace of a space X is paracompact if and only if every open subspace is paracompact. App. I PARACOMPACT SPACES 205 Proof. Let X be the space, let A be an arbitrary subspace, and let '11 = {U a} be an open covering of A. Extend each U a to a set U~ open in X, and let A'= Ua U~. This is an open subset of X, and therefore paracompact by hypothesis. Hence the open cover {U~} of A' has a subordinated partition of unity which restricts on A to one subordinated to '11. Since X is Hausdorff, the subspace A is also. The converse is trivial. I LEMMA 9. Let U = Ui:2:0 K i, where each Ki is a closed subset of the interior of Ki+l, and suppose each Ki is paracompact.