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Finite Topological Spaces FINITE TOPOLOGICAL SPACES BY R. E. STONG 1. Introduction. The object of this paper is to consider finite topological spaces: i.e. spaces having only a finite number of points. Most of the results obtained are clearly valid for spaces having only a finite number of open sets. In §2, an analysis of the homeomorphism classification of finite spaces is made and a representation of these spaces as certain classes of matrices is obtained. In §3, the point-set topological properties of finite spaces are considered. The topics covered are separation axioms, connectivity, mappings, and function spaces. In §4, the classification of finite spaces by homotopy type is made, reducing homotopy problems back to related homeomorphism problems. In §5, a clas- sification of finite //-spaces is made. It is shown that all such are of the obvious types. In §6, the homotopy classes of mappings of a finite simplicial complex into a finite space are examined. General references and definitions may be found in [2], [4], and [5]. The author is greatly indebted to Professor I. M. James for suggesting these questions and for discussions on these topics. The author is also indebted to the National Science Foundation for financial support during this work. 2. Homeomorphism classification. Proposition 1. Let F be a finite topological space with topology 3~. There exists a unique minimal base °lt for the topology. Proof. For each xeF, let Ux be the intersection of all open sets of F which contain x. Since F is finite, this is a finite intersection and so Ux is open. Let °U be the collection of all Ux. It is immediate that ^l' is a base for 3" and that any base for i?~ contains tft. Note. Spaces having this property, i.e. that any intersection of open sets is open, were considered by Alexandroff [1]. Proposition 2. Let F be a finite set, % a collection of nonempty subsets of F. Then 31 is the minimal base for a topology of F if and only if: il)\J{A\ AeK} = F, (2) if A,Be% there exist sets Cae3I such that AC\B = {JCa, and (3) // Axe 31 and {JXAXe % there exists a ß such that Aß = U<,4f Received by the editors July 16, 1965. 325 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 326 R. E. STONG [June Proof. Conditions 1 and 2 are the usual conditions that 31 is a base for a topology. If °U is the minimal base, then °U cz % Then if A e 51, A is a union of elements of °li since % is a base, but by condition 3 and Qlc%, A is one of the sets of °li of which it was the union. If °ll is a minimal base, (1) and (2) are trivial and if Ux = [J U¡, then there is a j such that x e Uj, so Ux c Uj by definition of Ux, so that (3) is satisfied. Proposition 3. Let F and G be finite spaces with minimal bases <?iand ir. Then : (a) J/ G is a subspace of' F,V' = {U n G | Ue°U}. iff) If F x G is the product space, then the minimal base is %x-r = {UxV\Ue<%,Ve r}. Definition. Let Ji' be the set of all square matrices (ai}) with integral entries such that: (l)««èl, (2) i#j=>a,7= -1,0, or 1, (3) i =¡é;=>«.,. : —ttji, and (4) for any sequence (¿i,i2,--,¿s), s > 2, ¿i t¿ i2 # •■• ^ fs such that If A,Be Ji' and there is a permutation matrix Twith T4T~' = B, then say A is equivalent to B. Let Ji be the set of equivalence classes of elements of Ji'. Theorem 1. The homeomorphism classes of finite topological spaces are in one-to-one correspondence with Ji. Proof. Let F be a finite topological space with minimal base ali. Choose an ordering of W so <%= {Uy,U2,---,Ur}. Define a matrix (au) i,j = !,•■■,r, by: Number of x e F 3 Ux = U¡ if j = i, 1 if U¡ =>Uj and 3 no ka U¡ =>Uk =>l/y -1 if Í/,- =>t/; and 3 no ka U¡ =>Uk zo U¡ if j # i. 0 otherwise Trivial verification shows that (au)eJ/'. Choosing a different order on % gives a permutation of {1, ••■,!•}and hence F determines an element of Ji. If f:F->G is a homeomorphism, then / induces a map of minimal bases, which preserves inclusions and numbers of elements in each base set, so F and G determine the same element of Ji. If F and G give the same element of Ji then by proper ordering of bases, they give the same matrix. A homeomorphism is easily constructed, thus showing that the function to Ji is monic. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1966] FINITE TOPOLOGICAL SPACES 327 To see that the function to J( is onto, let A = (a,,)e J£', with A an r x r matrix. Let S = {(a,b)eZ x Z\ 1 = a = r,l = b = aaa} and for i = l,»»»,r, let [(a,b)eS a = i or a i= i and 3 chain (iit ••», is), U,= i = ¿i ?»■¿2 # — # ¡5 = a with aili2 = ■•■= a. J- It is easy to verify that the collection of U, is a minimal base for a topology on S, and that with this topology S gives the matrix A. It should be noted that the size (r x r) of the matrix is the number of sets in the minimal base and that the trace ( Eai;) of the matrix is the number of points in the space. 3. Point-set topology. First, it is clear that if a finite space is T, then it is in fact discrete. However, T0-spaces are more useful and one has: Proposition 4 (Alexandroff [1]). A finite space F is T0 if and only ifUx= Uy implies x = y. Note. Thus F is T0 if and only if its matrix has only l's on the diagonal. Proof. If F is T0 and x ^ y, there is an open set V such that xeV, y $ V (or y e V, x $ V). Thus y f Ux (or x £ Uf) so Ux # Ur Conversely, if Ux = Uy implies x = y and u,veF such that every open set containing one contains the other, then v e Uu, u e Uv, so Uu = Uv and u = v. For T0-spaces, the partial order by inclusion on <?/gives a partial order on the space, and thus x — y is equivalent to Ux c Uy or xe Uy. For non-T0-spaces, one may also write x ^ y for x e [/,,, and the proposition then becomes : F is T0 if and only if x ^ y and y ^ x implies y = x. Now, consider connectivity. First, each set Ux is connected, for if A,B are open, Ux ezzA\j B, then x is in one set, say xeA and so Ux ezzA. Thus a finite space is locally connected. The usual open-closed argument shows: Proposition 5. Let F be a connected finite space, and x and y in F. Then there exists a sequence z0,z¡, -■■,zsof elements of F such that z0 = x, zs = y ami for each i (0—i = s—l), either z,-Sjz;+1 or z, — z,+ i. Proposition 6. Let F be a finite space, x,yeF and x = y. Then there is a path in F whose ends are x and y. Proof. Let ( y t = o 0:/ = [O,l]-+Fby eb(t)= \ I x í > 0 If V is any open set in F and (1) veF,thenx^vsoxe[/,c V and eb~xCV)= I. (2) x e V, y i V, then eb' l(F) = (0,1]. (3) x$V,thene¡>-\V)=0. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 328 R. E. STONG [June Thus cb is continuous. Combining Propositions 5 and 6, one has immediately: Corollary 1. Connectedness and path-connectedness are equivalent for finite spaces. Next, consider functions. Proposition 7. Let F and G be finite spaces, f : F-*G a function, f is con- tinuous if and only if x t%y implies f(x) ^f(y). Proof Iff is continuous, x £j y, then U f(y) is open so/_1(C/y(y)) is open and contains y. Thus it contains Uy and hence x (x e Uy). Thus f(x) e Uf(y) or f(x) gj/(y). Conversely, suppose x £j y implies/(x) gj/(y), and let Kbe open in G. If y ef~ \V), f(y) e V so UfW c V. Then if x e Uy, x = y and so f(x) £f(y) or f(x)eUf(y). Thusf(x)eVot xefV). Since f~l(V) = \Jyef.¡(v)Uy, it is open and so / is continuous. Proposition 8. Let F be a finite space, f a continuous map of F into itself. Iff is either one-to-one or onto, then it is a homeomorphism. Proof. Since F is finite, one-to-one and onto are equivalent. Since / is one- to-one, A-*f(A) defines a 1-1 correspondence/': 2*->2F. lff'(A) belongs to the topology &~ cz 2F,f(A) is open and by continuity and the 1-1 nature off, Ae3~. Since &~ is finite and ffT) =>f,f gives a 1-1 correspondence S"-*ïé'.
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