Turing Computability of (Non-)Linear Optimization Vasco Brattka Theoretical Computer Science I FernUniversit¨at Hagen 58084 GERMANY
Martin Ziegler Mathematics and Computer Science Universit¨at Paderborn 33095 GERMANY
[email protected]
[email protected]
Abstract
It is straight forward to adopt this model to Computational Geometry. In fact [6] already did so in investigating Turing computability of the extreme points of a convex body, a continuous generalization of classical convex hull problems. [11] considers geometric predicates on convex bodies such as M EMBERSHIP and L INE I NTERSECT in exactly the same sense. Also Gr¨otschel, Lov´asz, and Schrijver in their enlightening book [7] developed polynomialtime Turing-algorithms for C ONVEX O PTIMIZATION up to prescribable error " > 0: although not mentioned explicitly, their W EAK O PTIMIZATION P ROBLEM is exactly in the spirit of Recursive Analysis. The present work generalizes a fundamental result from [7]: Dropping the requirement of polynomial time complexity, we present Turing algorithms for the optimization of arbitrary continuous objective functions. Even the convexity condition may be relaxed: It is shown that one can computably optimize over compact, connected, and everywhere full-dimensional domains, so-called ‘bodies’. To this end, we establish new computability results for operators on bodies such as union, intersection, and function pre-image. The latter considerations resemble similar ones in [5] which, however, were performed in the domain-theoretic setting.
We consider the classical L INEAR O PTIMIZATION problem, but in the Turing rather than the REAL -RAM model. Asking for mere computability of a function’s maximum over some closed domain, we show that the common presumptions ‘full-dimensional’ and ‘bounded’ in fact cannot be omitted: The sound framework of Recursive Analysis enables us to rigorously prove this folkloristic observation! On the other hand, convexity of this domain may be weakened to connectedness, and even non-linear functions turn out to be effectively optimizable.
1 Motivation The gap between (theoretical) algorithm design and (practical) implementation reveals a particular challenge in Computational Geometry: Provably correct algorithms keep, upon implementation, reporting not only inaccurate but sometimes even entirely invalid results, especially upon input of degenerate configurations [9] — for obvious reasons: Algorithms in Computational Geometry are most generally developed in the REAL -RAM model [1], capable of operating on real numbers exactly. Actual digital computers however can process in each step only a finite amount of information [2]. We therefore find it necessary to consider a different model of real number computation. In fact, Alan Turing himself introduced ’his’ machine in order to study computability aspects over R [14] and thereby initiated the nowadays well-established field of R ECURSIVE A NAL YSIS [8, 12, 15]. Roughly speaking, a Turing machine is said to compute the real number r if it can output rational approximations of arbitrarily prescribable precision.
2
Introduction
Recursive Analysis combines Classical Analysis with discrete Recursion Theory: Results of the former such as I N TERMEDIATE VALUE or F IXED P OINT Theorems claiming existence of certain objects are investigated with respect to the computability of these objects [13, 10, 16, 18]. A real number r 2 R for example is computable if there exists some Turing machine which outputs two sequences pn and qn of integers such that jr - pn =qn j < 2-n .
Partially supported by DFG Grants Me872/7-3 and Br1807/4-1
1
f : X ! Y is ( ; )-computable, then f is continuous: see Theorem 3.2.11 in [15]. The issue of non-/uniform computability is best illustrated by comparing the following two statements:
Obviously, every rational number is computablepin this sense; and so are some irrationals such as , e, 2. On the other hand not every of the uncountable many real numbers can be computed by only countable many Turing machines: any non-recursive subset N of N gives rise P to a non-computable real n2N 2-n . As usual, Turing machines become able to compute nonnumeric objects (e.g., graphs) by encoding them into binary strings and operating on these ‘names’: Following the general framework of T YPE -2 T HEORY OF E FFEC TIVITY [15], let X be an arbitrary set not exceeding continuum cardinality jXj and : N ! X a fixed partial surjective mapping, a so-called representation. Machine M is said to -compute x 2 X if it outputs an -name for x, that is, an infinite string ¯ 2 N such that (¯ ) = x. For infinite outputs to make sense after finite time, the output tape is required to be one-way. In the above example X = R, let ( ¯ ) be defined for all strings ofthe form ¯ = bin(p1 ), bin(q1 ), bin(p2 ), bin(q2 ), : : : where r := limn!1 (pn =qn ) exists and -n jr - pn =qn j < 2 ; here bin(i) denotes usual binary encoding of integer i. This : ¯ 7! r is the standard representation of real numbers. Different representations of the same set X may induce different notions of computability. For instance define < (¯ ) to be the real number r iff ¯ is (an encoding of) a sequence of rationals converging to r from below. It is easy to see that any -computable real is < -computable as well, but not vice versa. If however some r 2 R is both < -computable and > -computable — the latter defined analogously by convergence from above — then r must be -computable, too. The preceding considerations dealt with computability of single objects x. This was reflected by the fact that the corresponding machine M received no input but had to output a name for x ‘from scratch’. Let us now fix two sets X and Y with corresponding representations and . A function f : X ! Y is ( ; )-computable if some Turing machine, upon input of any -name ¯ for x 2 X, outputs a -name ¯ for y = f(x). The M AIN T HEOREM OF R ECURSIVE A NALYSIS now states that any (; )-computable real function must necessarily be continuous! More generally: Let N be equipped with the Cantor Topology and : N ! X, : N ! Y admissible representations, (i.e., ‘wellbehaved’ w.r.t. the final topologies on X and Y ). If
x -computable ) f(x) -computable. Effectivity: f : x 7! f(x) is ( ; )-computable. Call a representation of X reducible to 0 of X ( 0 ), if some Turing machine converts -names into 0names for the same object x 2 X. Equivalent, i.e., pairHeredity:
wise reducible representations lead to identical notions of uniform computability. It holds, for example,
< ; > ; < 6 ; > 6 ; (< + > ) We mention two ways of constructing new representations from given ones, details can be found in [15]: Let 1 ; : : : ; n be representations of X 1 ; : : : ; Xn , respectively. Qn There exists Qna canonical representation, denoted i=1 i , of i=1 Xi .
3
Let and be admissible representations of X and Y , respectively. There exists a canonical representation of C(X; Y ), the space of continuous functions f : X ! Y , which will be denoted by [ ! ℄.
Computability of Sets
For subsets of Rn , several notions of computability are treated in literature. For instance, ; 6= A Rn is Turinglocated [6] iff the Euclidean distance function
R
dA : n
! R;
x 7! ainf 2A
r
Xn i=1
xi - ai )2
(
is (n ; )-computable. This d A can be regarded as continuous analogue to the characteristic function of A which, except for trivial cases A = ; and A = R n , is never computable nor continuous. It is easy to see that dA characterizes the set A uniquely, provided A is closed. In other words, we have a representation (from now on called n ) of the hyperspace A n of closed non-empty subsets of R n : ¯ is a n -name for A iff ¯ is a [n ! ℄-name for dA . On the other hand, no representation of all subsets can exist as there are too many of them: j2 Rj > . By definition, A 2 An is Turing-located iff it is n n computable. Two weaker notions, n < and > , ask for n n [ ! > ℄- and [ ! < ℄-computability of d A , respecn n tively. Notice the reversed index: > = [ !< ℄. 2
For compact, i.e. bounded closed, sets it turns out that many problems become computable only if, in addition to the n -information on A, some explicit upper bound on its diameter is supplied. Let
n n K := (A; r) : A 2 A ; r 2
N; A
n [-r; r℄
Another problem arises from the fact that the hyperspace of compact subsets of R n does not contain halfspaces nor is it effectively closed under intersection or pre-image [3]: – Binary union [ : A n An ! An , (A; B) 7! A [ B is ( n n ; n )- and (n n ; n )-computable; – whereas intersection (A; B) 7! A \ B is not even (n n ; n < )-continuous. – Function pre-image A 7! f -1 [A℄ is not (m ; n < )computable, even for computable f : R n ! Rm .
N
n A
denote the hyperspace of all non-empty compact subsets and n := ( n bin) its canonical representation; simin n larly n < and > . We tacitly assume that K be embedded n into A the obvious way. Non-uniform it holds that compact A Rn is n -computable iff it is n -computable. n Kn n K 6 n . Also, obviously n n ; but It is well-known that n (but not n ) permits a computable version of the famous H EINE -B OREL Theorem, see [15]. Even more, the maximum of arbitrary continuous functions over compact sets can effectively be computed:
4
It is important to observe that the two-dimensional Example 2 heavily relies on the set L(A; b) to become onedimensional for " = 0. In fact, we track down the (non-) computability of L INEAR O PTIMIZATION to an issue of full-dimensionality rather than non-degeneracy. More precisely, our new Theorem 4 shows that operations union, intersection, and pre-image become effective when restricting to everywhere full-dimensional sets. Call set A everywhere full-dimensional (or regular) if it coincides with the closure of its topological interior A Æ . This notion generalizes the full-dimensionality from convex sets, cf. Lemma 11 of the appendix. A solid is a regular non-empty proper subset of R n . The below figure exemplifies, in dimension 2,
Lemma 1 (Corollary 6.2.5 in [15]) Maximum of continuous functions over a compact set C(R n; R) Kn 3 (f; K) 7! max f[K℄ 2 R is [n!℄ n ; -computable. On the other hand, compact sets are quite often described not by n -names but, for example a convex body, as solutions to a system of linear inequalities:
L(A; b) = x 2 Rn : A x b ; A 2 Rmn ; b 2 Rm The very important question is thus: Can ( mn m )names for (A; b) be converted to n -names for L(A; b)?
A ) A SOLID
In general, they can’t. Even more, L INEAR O PTIMIZA TION with degenerate constraint hyperplane configurations is in general not computable! This folkloristic observation, usually avoided by general position presumptions, now follows rigorously in the sound framework of Recursive Analysis from its Main Theorem and the following
"x 0 -1 1
0
i.e.
" B 0 A=B -1 1
L: 9 A 2 Xn ()
1
d)+h)
0 1
" > 0: L(A; b) = convHull (0; 0); (-1; 0); (-1; -") f[L(A; b)℄ = [-1; 0℄, max f[L(A; b)℄ = 0 " = 0: L(A; b) = [-1; 1℄f0g, max f[L(A; b)℄ = 1
B ) A NON - SOLID SET.
;= 6
A = Rn n B
for
;= 6
B := Rn n A
Encode A 2 X simultaneously by n > -names for both A and B, i.e., effective approximations from below of two distances: to the set and to its closed complement. Let n denote the corresponding representation. On the other hand by virtue of Lemma 10d):
1 0 B0C 1C C B C 0A b = 1 A 0 1
-
AND
Let Xn denote the hyperspace of all solids in R n . It holds:
Example 2 (Discontinuous LP Optimum) In dimension 2, fix f : (x; y) 7! x and 0 " -name for A and a > -name for any B with 3
A = Rn n B. Let n denote this joint representation. We 5
Conclusion
made the important observation that both approaches are only logically but not computationally equivalent:
We presented Turing-computability results for BODIES, i.e., non-empty bounded connected regular closed subsets of Rn . As objects cannot be physically distinLemma 3 n n , but n 6 n . guished anyway if they differ only on a nowhere dense It turns out that n serves better for generating n set, the full-dimensionality requirement is no restriction in names of L(A; b): Many so-far discontinuous operators practice. Theorem 5 generalizes previous considerations on closed sets become n -computable after restricting [4] on computability of linear equalities to inequalities. them to solids. For intersection and pre-image, precauProof techniques however are entirely different: Topoltions have to be taken to avoid non-solid results: ogy rather than Linear Algebra, see Appendix A and B. Theorem 4 a) Union is ( n n ; n )-computable.
References
b) Intersection of solids set is (n n ; n )computable whenever well-defined, i.e., on (A; B) :
A; B 2
n n X ;A\ B 2 X
n
n;
[1] M. de Berg, M. van Kreveld, M. Overmars, O. Schwarzkopf: ”Computational Geometry”, Springer (1997). [2] V. Brattka, P. Hertling: ”Feasible Real Random Access Machines”, pp.490-526 in Journal of Complexity 14 (1998). [3] V. Brattka, K. Weihrauch: ”Computability on Subsets of Euclidean Space I: Closed and Compact Subsets”, pp.65-93 in Theoretical Computer Science 219 (1999). [4] V. Brattka, M. Ziegler: ”Computability of Linear Equations”, submitted; further information available from http://www.phys.upb.de/˜ziegler/basis.html [5] Abbas Edalat, A. Lieutier: ”Foundation of a Computable Solid Modelling”, to appear in Theoretical Computer Science (2002). [6] Xiaolin Ge, Anil Nerode: ”On extreme points of convex compact Turing located sets”, pp.114-128 in Logical Foundations of Computer Science, Springer LNCS 813 (1994). [7] M. Gr¨otschel, L. Lov´asz, A. Schrijver: ”Geometric Algorithms and Combinatorial Optimization”, Springer (1988). [8] A. Grzegorczyk: ”On the definitions of computable real continuous functions”, Fundamenta Mathematicae 44 (1957) 61-77. [9] P. Hertling, K. Weihrauch: ”Levels of Degeneracy and Exact Lower Complexity Bounds for Geometric Algorithms”, pp.237242 in Proceedings of the 6th CCCG (1994). [10] Ker-I Ko: ”Complexity Theory of Real Functions”, Progress in Theoretical Computer Science, Birkh¨a user Boston (1991). [11] M. Kummer, M. Sch¨afer: ”Computability of Convex Sets”, pp.550-561 in Proceedings of the STACS, Lecture Notes in Computer Science 900, Springer (1995). [12] Daniel Lacombe: ”Les ensembles r´ecursivement ouverts ou ferm´es, et leurs applications a` l’Analyse r´ecursive”, pp.28-31 in Compt. Rend. Acad. des Sci. Paris, 246 (1958). [13] M.B. Pour-El and J.I. Richards: ”Computability in Analysis and Physics”, Springer (1989). [14] Alan M. Turing: ”On computable numbers, with an application to the Entscheidungsproblem”, pp.230-265 in Proc. of the London Math. Soc. 42(2) (1936). [15] Klaus Weihrauch: ”Computable Analysis”, Springer (2000). [16] Kam-Chau Wong: ”Computability of Minimizers and Seperating Hyperplanes”, pp.564-568 in Math. Logic Quaterly 42 (1996). [17] M. Ziegler, V. Brattka: ”Computing the Dimension of Linear Subspaces”, pp.450-458 in SOFSEM’2000: Theory and Practice of Informatics, Springer LNCS 1963 (2000). [18] M. Ziegler, V. Brattka: ”A Computable Spectral Theorem”, pp.378-388 in Computability and Complexity in Analysis, Lecture Notes in Computer Science 2064, Springer (2001).
n Xn X
c) Pre-image under continuous open mappings f 2 C(Rn ; Rm ) is [n!m ℄ m ; n -computable. d)
even restricted to solids,
n jXn
6
n .
e) Restricted to non-empty, compact and connected n n fA2K connectedg n sets, Combining parts d) and e) and Lemma 3 with Lemma 1, it immediately follows that any computable function can effectively be maximized over a body (compact, connected, non-empty, everywhere full-dimensional) B R n , given by either a n -name, a n -name, a n -name, or a n name! Let us now show how such a name for full-dimensional L(A; b) can be obtained from a ( mn m )-name of (A; b): For 0 2 2 Rn , H0 ; denotes the R, 0 n6= P closed halfspace x 2 R : n i=1 i xi 0 .
3 a) The mapping R Rn n f0g n (0 ; ) 7! H0 ; 2 X is ( n ; n )-computable.
Theorem 5
b) Upon (mn m )-input of (A; b) 2 Rmn Rm , L(A; b) is n -computable, provided A contains no zero-rows and L(A; b) is full-dimensional. Let us remark that, apart from full-dimensionality (Example 2), boundedness is an essential prerequisite, too, in order to ensure computability of L INEAR O PTIMIZATION: Example 6 In dimension 2, fix f : (x; y) 7! x. The LP
x "y
A = (1; -"); b = (0) defines, for each 0 " < 1, a full-dimensional domain L(A; b). But max f[L(A; b)℄ is discontinuous in ": = 0 for " = 0. max f L(A; b) = 1 for " > 0, i.e.
4
A
Proofs
i) Type-conversion of functions is effective, that is, we have UTM- and SMN-like properties: Let , , be admissible representations for X,Y , and Z, respectively. Then
Here come the proofs which had so far been omitted due to space limitations. Relying on previous results on setcomputability in [3, 15], we will need one additional representation namely of the hyperspace O n of open (possin bly empty) subsets U R n : A n < -name for U 2 O is a sequence
x i ; ri ) 2 Q n Q + ; i 2 N ;
(
s.t.
U=
[
X C(X Y; Z) 3 (x; f) 7! f(x; ) 2 C(Y; Z) is [ ! ℄; [ ! ℄ -computable. Proof: See Theorem 5.1.13, Exercise 5.1.21, Theorem 6.2.4, Theorem 3.3.15 in [15].
B(xi ; ri )
i2N Similarly, a n > -name for U is a list of all (!) rational (x; r) such that B(x; r) \ (Rn n U) 6= ;. Let us point n n , n , n , n can out that representations n >, easily be extended to encode also empty sets [3]; the trick is to permit 1 as possible value of the distance function dA , i.e., consider [n !¯ ℄-names where ¯ is a canonical extension of to R = R [ f1g.
In some of the following proofs, we shall deliberately refer to facts from Topology which are collected in Appendix B. Proof: [Lemma 3] The first claim is trivial, because any n -name for A 2 Xn is a n -name for A as well. We will now show that n -names for solids cannot in general depend continuously on their respective n -names. Let us consider the one-dimensional case:
Let us now recall some well-known computability results: Lemma 7 (Toolbox)
0 0 n is both 3 (A; A ) 7! A [ A 2 A n n n n n n ( > > ; > )- and ( < < ; < )-computable. d) Intersection An An 3 (A; A 0 ) 7! A \ A 0 2 An is n n n n n ( n > > ; > )- and (> > ; > )-computable.
U := (-1; -") [ (+"; +1); A := U; 0" -names for A and B can be chosen which depends continuously on ". So we have, for each " 0, a -name for A. But a -name must contain > -information about B 0 := R n A rather than B; and for " > 0, B 0 = B = (-1; -1℄ [ [-"; +"℄ [ [+1; +1) whereas for " = 0, B 0 = (-1; -1℄ [ [+1; +1): The function " 7! dB (0) ‘jumps’ from 0 to 1 as " & 0 and
e) Pre-image
thus cannot be (lower semi-) continuous.
a) Complement On 3 U 7! Rn n U 2 n n n ( n < ; > )- and ( > ; < )-computable. b) Complement An 3 A 7! Rn n A 2 n n n (n < ; > )- and (> ; < )-computable.
A
n is both
O
n is both
c) Union An An
R R
C( n ; m ) Am 3 (f; A) 7! f-1 [A℄ n is [n!m ℄ m > ; > -computable.
0
n 2 A
Proof: [Theorem 4a)] Let A; A 0 2 Xn , A = Rn n B and A 0 = Rn n B 0 for B; B 0 2 An . Keep in mind that a n name for A consists of n > -information about both A and B; analogously for A 0 ; B 0 . By virtue of Lemma 7c), we can effectively obtain n >information on A 00 = A [ A 0 . Let us also compute n >information on B 00 := B \ B 0 according to Lemma 7d).
f) Pre-image is
C(Rn ; Rm ) Om 3 (f; A) 7! f-1 [A℄ 2 On n [n!m ℄ m < ; < -computable.
g) Closure On computable.
3
U ! 7 U
2
A
n is (n ; n )<
-information for A 00 = A [ A 0 and for some closed B 00 with Rn n B 00 = A 00 which, according to the definition of n , is sufficient. This also implies regularity, i.e., A 00 2 Xn . [
h) Among compact sets, the property of being empty is n> -r.e., in other words: Upon n> -input of compact K 2 Kn , a Turing machine can in finite time verify (but not decide) whether K = ;. 5
=
Other than union, the intersection of regular closed sets is Proof: [Theorem 4d)] Let A and B be given by their n is equivalent to in general not regular again, consider A = [-1; 0℄ and respective n > -information. Since 0 0 n n n n A = [0; 1℄. The requirement that A \ A 2 X is the join of < and >, we need < -information about therefore necessary for its n -computability. We will now A = Rn n B. Employ Lemma 7b) and g) to compute it. show, that it is even sufficient: Proof: [Theorem 4b)] Let A; A 0 ; A 00 2 Xn , A 00 = A \ A 0 . By definition, A = Rn n B, A 0 = Rn n B 0 and A 00 = Rn n B 00 for closed sets B; B 0 ; B 00 . 00 A n > -name for A is easily obtained from those of A 0 and A by virtue of Lemma 7d). It is in general not pos00 sible to compute a n > -name for B . On the other hand, n -information about any closed C with A 00 = Rn n C > suffices. And C := B [ B 0 is easily computable, see Lemma 7c). It thus remains to show R n n C = A 00 . To this end, let us apply Lemma 12b). Set W := R n n B 00 , U1 := Rn n B, U2 := Rn n B 0 . By presumption, W = A 00 = A \ A 0 = U1 \ U2 , hence
A 00 = W
L:12 =
Proof: [Theorem 4e)] Let connected A 2 K n be given by a n -name. Determine, according to Exercise 5.1.13b in [15], some x 2 A 6= ;. The mapping r 7! B(x; r) is obviously (bin; n )-computable. By virtue of Lemma 7d) and h), we can verify, for each r 2 N , whether A \ B(x; r) = ;. As A is compact, such an r exists; and using dove-tailing, we can effectively find one. Now r + kxk2 is a bound for A, because A is connected to x. Proof: [Theorem 5] For 0 consider the affine function
U1 \ U 2 = R n n C
f : Rn ! R;
according to de Morgan’s law.
2
x 7!
R and nonzero Rn , 2
Xn
x - 0 i=1 i i
:
By Lemma 7i), f is [ n !℄-computable upon ( n )input of (0 ; ). Also in finite dimension, a linear funcAlso for pre-image, the requirement of f being open is tion is open iff it is surjective. As 6= 0, this is the essential: case for f. And the constant regular set [0; 1) is triv2 2 ially By virtue of Theorem 4c), H 0 ; = Example 8 (non-open pre-image) Let f : R ! R , -1 -computable. n -computable. is f [ 0; 1 ) (x; y) 7! (x; 0) and A" := ([-1; 0℄ [-1; +1℄) [ ([0; 1℄ ["; 1℄). This f is continuous but not open; A " a con- For part b), the affine function Xn nected solid for any 0 " 1. It is easy to see that g : R n ! Rm ; x 7! i=1 A x - b n -information can be presented in such a way that it depends continuously on ". Also, B " := f-1 [A" ℄ is a con- is not open for m > n. Instead, let a i 2 Rn denote the i-th row of A, i = 1 : : : m. Since ai 6= 0, all halfspaces nected solid for any ": Hi = Hbi ;ai are n -computable according to a).TAs long B0 = [-1; +1℄ R; B" = [-1; 0℄ R; " > 0 as the results are regular, intersection L(A; b) = m i=1 Hi n However, -information about B " cannot depend con- is n -computable: Theorem 4b). tinuously on and therefore not on A " , either. Proof: [Theorem 4c)] Let A 2 X m , A = Rm 0 -1 Lemma 7e) to n > -compute A = f [A℄ 0 -1 n B := f [B℄ R . Then,
Rn n B 0 = f-1 [Rm n B℄
L:13 = a)+b)
f-1
h
Rm n B
n
B
B Use
Rn
i =
Topology
This section contains some topological lemmas which serve as ‘tools’ for the above proofs. From the many equivalent categorial approaches to set theoretic Topology, we prefer the following one: A topological space is a set X together with a collection O f;; Xg of subsets of X, closed under finite intersections and arbitrary unions. Members U 2 O will be called
and
A0
as f is both continuous and open.
6
open, complements X n U are closed. For M X, let
M :=
\
MAX A closed
A
Æ M :=
and
[
UM U open
C. Based on the indepence of the b i , it is easy to see that already S has positive volume and non-empty
U
interior. Lemma 12 a) Let U X be open, Q but dense in X. Then U \ Q = U.
denote its closure and interior, respectively. The following facts are straight forward to verify:
b) Let U1 ; U2 ; W X open and U 1 \ U2 Then W = U1 \ U2 .
I denote an arbitray index set and G; H;[ Mi X, i [ 2 I. \ Æ Æ \ Æ Æ a) e) Mi Mi Mi Mi i2I i2I i2I i2I
Lemma 9 Let
\
\
c)
Mi Mi GÆ \ HÆ = (G \ H)Æ
g)
d)
(
h)
b)
X n G)Æ = X n G
f)
[
Mi
[
X arbitrary =
W.
c) Let XTbe a Baire space, U n ; W TX open for n 2 N and n2N Un = W. Then W = n Un .
Proof: a) U \ Q U yields U \ Q U by monotonicity. For the reversed inequality let A := U \ Q. The closed set A 0 := A [ (X n U) contains (Q \ U) [ (Q n U) = Q so, again by monotonicity, A 0 = A 0 Q = X. But A 0 = X implies U A = U \ Q.
Mi
G[H=G[H X n H = X n HÆ
b) Let us first treat the case W = X: Setting U := U1 and Q := U2 , part a) gives the desired claim which is, in words: The intersection of two open dense subsets is dense again.
Lemma 10 Let U; V; W and A; B; C denote, respectively, open and closed subsets of X.
Æ UU A AÆ
Æ U=U AÆ = A
Lemma 11 Let C Rn be non-empty, closed, and convex. The following are equivalent: b) C Æ 6= ; c) CÆ = C a) dim(C) = n
For W ( X, let Ui0 := Ui [ (X n W): These sets are open and dense in X according to Lemma 9g); hence U10 \ U20 is, too: the case just treated. One may thus let Q := U10 \ U20 and apply a) once again to obtain W \ Q = W. Verifying W \ Q = W \ U1 \ U2 , it follows U1 \ U2 W; the other inclusion holds by virtue of Lemma 9b).
C =) CÆ , thus ; 6= CÆ . b)) c) Fix p 2 CÆ and r > 0 such that B(p; r) C. Let x 2 C, we show x 2 CÆ : This proves C CÆ , the
c) Recall that in a Baire space by definition, the countable intersection of open dense subsets is dense again. This proves the case W = X from which the case W ( X follows as above.
a) b)
Proof: c)) b)
c) d)
U = AÆ A=U
) )
; 6=
other inclusion follows from Lemma 10b). W.l.o.g. x = 0, otherwise consider C 0 := -x + Lemma 13 Let X; Y be topological spaces, f : X ! Y , C. By convexity, x n := p=n 2 C. Even more, G X, H Y . B(xn ; r=n) C. Thus xn 2 CÆ for all n 2 N , a) If f is continuous, then hence x = lim xn 2 CÆ . h i hÆi h i f G f[G℄ f-1 H f-1 [H℄ f-1 H f-1 [H℄ Æ b)) a) Let B(p; r) C Rn . Then by monotonicity,
n = dim B(p; r) dim C dim Rn
=
n:
b) If f is open, then
hÆi hÆi h i f G f[G℄ Æ f-1 H f-1 [H℄ f-1 H f-1 [H℄ Æ
a)) b) By presumption, the affine hull of C spans whole Rn . W.l.o.g. 0 2 C, i.e., one may consider the linear rather than affine hull. Choose a basis b 1 ; : : : ; bn 2 C for Rn and let b0 := 0 2 C. By convexity, the simplex S spanned by b 0 ; b1 ; : : : ; bn is contained in
c) If f is closed, then
7
h i
fG
f[G℄.
