DEMONSTRATIO MATHEMATICA Vol. XXI No J 1988 Jarostaw Wröblewski SINGULAR FOLIATIONS OF DIFFERENTIAL SPACES 1. Introduction The notion of the nonsingular foliation of a differen- tiabie manifold was introduced by ¿'hresmann and Reeb in [2]. In [8] and [9] Stefan generalized that notion and introduced the notion of the foliation with singularities. In [11] Wali- szewski introduced the concept of nonsingular foliation in the category of Sikorski differential spaces (see [3], [6] and [7]). In the present paper we introduce the ootion of singular foliations of differential spaces and Btudy some properties of this notion. For definition of differential spaces we refer to the papers [1 ], [3], [4], [6], [7], [11], [12] and [14]. We introduce the following notational conventions used in the present paper. Let U be a differential space or a topological space. Then we denote: setM - the set of all points of U, topM - the topology of M, Ma for A c setM - the subspace of U with the set of all points A. For a differential space M we put TopM = (setM, topM). The distinction of a space from its set of all points makes possible the following convention. By f:M —»• N we mean that f is a morphism in a suitable category, namely - 805 - 2 J. V/roblewski - f is a smooth map of K into N, if M and K are diffe- rential spaces, - f is a continuous map of M into N, if Ivl and N denote topological spaces, - f is any set-theoretical function from K into K, if M and E are sets. By an imbedded differential subspace of a differential space Ivl we mean any differential space K with setK csetM such that for any point p of N there is such an open in topK neighbourhood U of p that Ny = My. It is obvious that if K and N are imbedded differential subspaces of M with TopK = = TopK then K = N. 1.1. Definition. Let M be a differential space and let X be a topological space with setX csetM and idsetx:X —TopM. Let C be the class of all such functions (3:setX —»• that for any p setX there exist a neighbourhood U of p open in X and a smooth real function a on M satisfying the condition p|y = °<|u* Then C is a differential structure on setX. The differential space (setX,C) we denote by M^. We have TopMx = X. Any space of the form Mx will be called a pre-imbedded differential subspace of M. Any imbedded sub- space is a pre-imbedded subspace. 2. Definition of foliation 2.1. Definition. Let p be a point of a dif- ferential space M. By a looal trivialization of M around p we mean any system (B,T,U,f), where B, T and U are subsets of set&I containing p, U is open in topM and f is such a func- tion f :B*T —• U that f(p,.) = idT, f(.,p) = idg and f maps diffeomorphically MgXMj onto My. 2.2. Definition. Bya foliation on a dif- ferential space M we mean a differential space F such that setM = setP and for any point p of M there is such a local trivialization (B,T,U,f) of M around p that B is open in topF, PN = MB and (B,T,U,f) is a local trivialization of F around p. We say that (B,T,U,f) is a local trivialization distinguished by P. - 806 - Singular foliations 3 2.3. Corollary. Any foliation ? of M is re- gularly lying (sea [12]) in M. 2.4. Corollary. Any foliation F of M is an imbedded differential subspace of M. 2.5. Corollary. If fi and F are foliations of M and TopS = TopP then S = P. 2.6. Definition. By a leaf of a foliation F of ivi we mean any connected component of TopP. Following Tamura [10] we prove the following topological properties of leaves. 2.7. Theorem. Let F be a locally connected foliation on M and let A be an union of leaves. Then the closure A in TopM is on union of leaves too. Proof. We have to prove that Ic A for any leaf L intersecting A. Let us take any point p of L and a triviali- zation (B,T,U,f) distinguished by F around p with connected PQ. Then for any point s of T the set f(B*{s}) is contained in a leaf. Since then we have BcAorBnA = 0. Thus A is open and closed in F, hence is an union of leaves. The following two corollaries from Theorem 2.7 hold for locally connected foliations. 2.8. Corollary . For any U open in TopM the union W of all leaves intersecting 'J is open in topM. Proof. Let A = setM\W. Then A is an union of leaves. By Theorem 2.7 A is an union of leaves too and since A nU = 0 we have A = setM\W, what ends the proof. Let M/F denote the topological quotient space of TopM by the equivalence, relation determined by the decomposition of M into the leaves of F. 2.9. Corollary. The canonical projection from TopM onto M/F is open (cf. [5]). 3. Operations on foliations 3.1. Corollary. IfF and G are foliations on differential spaces M and N respectively then their pro- duct FxG is a foliation on M*N. 807 - 4 J. Wrdblewaki 3.2. Corollary . For any foliation F on U and any set U open in topM Fy is a foliation on My. 3.3. Corollary. If F is an imbedded differen- tial subspaoe of M with eetF = setU and for any point p of M there is a neibhbourhood U of p open in topM such that F^ is a foliation on M^ then F is a foliation on M. 3.4. Definition. We say that a point p is a nonsingular point of a foliation F on M if there exists a local trivialization (B,T,U,f) around p distinguished by F in with F^ being a discrete space, i.e. F^ = (T, IR ). 3.5. Corollary. The set of nonsingular points of F is open in topU. One may think that we oan introduce the notion of a non- singular leaf of a foliation. The following example shows that a leaf may contain singular and nonsingular points simultaneously. 3.6. Example. Let for a differential space K and itq subset S csetK, K^g denote the differential spaoe with setK^g = eetKx{l} u S * IN, IN being the set of all natu- ral numbers, and with the differential structure containing all functions of the form » o pr^, where cx is a smooth real function on K and pr^ denote the projection onto the first coordinate. Saying roughly: K^ g is the spaoe K, where any point of the set S is considered as infinite number of points 2 2 in tne same place. Put M' = (IR , oC°°(lR )) and let F be the differential space with setF'= IR and the differential 2 structure containing all real functions a on IR such that c>(.,x) e C°°(lR) for any real x. Then F' is the decomposition of the plane into the horizontal lines. Put S=(0,1 )xlR u {-oo,0]* x(IR \{o}). Then F = F' Q is a foliation on M = M' c. The set 00 i r i r l °° of all singular points of F is (-oo ,0Jx \0} »{1} and is con- tained in the leaf iR*{o} *{l} u (0,1) *{o}*iN, but not equal to it. 3.7. Definition. We say that a foliation F on M is nonsingular iff all points of F are nonsingular. Since Waliszewski (see [ll] ) assumed foliation to be locally homogeneous, the above definition of nonsingular - 808 - Singular foliations 5 foliation of differential space is not equivalent to that of Waliszewski. We notice that Corollaries 3.2 and 3.3 hold when the notion "foliation" is replaced by "nonsingular foliation". 3.8. Theorem. Let g:M —» N be any coregular mapping of differential spaces (see [12]). Let F be a diffe- rential space defined as follows: setF * setM and F is the disjoint sum of all spaces of the form M 1 , where q e setN. g (q) Then F is a nonsingular foliation on M. For the proof it sufficies to use the definition of co- regularity and apply Corollary 3.3 for nonsingular foliations. 4. Univeasal local connectedness 4.1. Definition. Let X be a topological space. By a topological universally locally connected (shortly: Ulc) subspace of X we mean a locally connected topological space Y with setY c setX such that id8ety:Y —• X and for any mapping f:T —of a topological locally connected space T we have f:T —»• Y. We do not assume Y be a topologi- cal subspace of X. 4.2. Corollary. If Y and Z are topological Ulc-subspaces of X and setY = setZ then Y = Z. Indeed, the mapping z is a homeomorphism. 4.3. Definition. By a differential Ulc-sub- space of a differential space M we mean any pre-imbedded subspace N such that TopN is a topological Ulc-subspace of TopM. 4.4. Corollary. For any differential Ulc-sub- spaces N and L of M with setN = setL we have N = L. Moreover, if M is a differential space and X is a topological Ulc-sub- space of TopM then there exists unique differential Ulc-sub- space N of M such that TopN = X, namely N = M^.