A COUNTABLE SPACE WITH AN UNCOUNTABLE FUNDAMENTAL GROUP

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1 A COUNTABLE SPACE WITH AN UNCOUNTABLE FUNDAMENTAL GROUP JEREMY BRAZAS AND LUIS MATOS Abstract. Traditioal examples of spaces that have ucoutable fudametal group (such as the Hawaiia earrig space) are path-coected compact metric spaces with ucoutably may poits. I this paper, we costruct a T 0 compact, path-coected, locally path-coected topological space H with coutably may poits but with a ucoutable fudametal group. The costructio of H, which we call the coarse Hawaiia earrig is based o the costructio of the usual Hawaiia earrig space H = 1 C where each circle C is replace with a copy of the four-poit fiite circle. 1. Itroductio Sice fudametal groups are defied i terms of maps from the uit iterval [0, 1], studets are ofte surprised to lear that spaces with fiitely may poits ca be path-coected ad have o-trivial fudametal groups. I fact, it has bee kow sice the 1960s that the homotopy theory of fiite spaces is quite rich [10, 14]. The algebraic topology has fiite topological spaces has gaied sigificat iterest sice Peter May s sequece of REU Notes [7, 8, 9]. See [2, 3, 5] ad the refereces therei for more recet theory ad applicatios. While it is reasoable to expect that all fiite coected spaces have fiitely geerated fudametal groups, it is rather remarkable that for every fiitely geerated group G oe ca costruct a fiite space X so that π 1 (X, x 0 ) G. I fact, every fiite simplicial complex is weakly homotopy equivalet to a fiite space [10]. I the same spirit, we cosider the cosider fudametal groups of spaces with coarse topologies. It is well-kow that there are coected, locally path-coected compact metric space whose fudametal groups are ucoutable [4]. Sice fiite spaces ca oly have fiitely geerated fudametal groups, we must exted our view to spaces with coutably may poits. I this paper, we prove the followig theorem. Theorem 1. There exists a coected, locally path-coected, compact, T 0 topological space H with coutably may poits such that π 1 (H, w 0 ) is ucoutable. To costruct such a space, we must cosider examples which are ot eve locally fiite, that is, spaces which have a poit such that every eighborhood of that poit cotais ifiitey may other poits. Additioally, sice our example must be path-coected, the followig lemma demads that such a space caot have the T 1 separatio axiom. Lemma 2. Every coutable T 1 space is totally path-discoected. Date: April 8,

2 2 JEREMY BRAZAS AND LUIS MATOS Proof. If X is coutable ad T 1 ad α : [0, 1] X is a o-costat path, the {α 1 (x) x X} is a o-trivial, coutable partitio of [0, 1] ito closed sets. However, it is a classical result i geeral topology that such a partitio of [0, 1] is impossible [12]. Ultimately, we costruct the space H by modelig the costructio of the traditioal Hawaiia earrig space H, which is the prototypical space that fails to admit a uiversal coverig space. The fudametal group of the Hawaiia earrig is a ucoutable group which plays a key role i Eda s homotopy classificatio of oe-dimesioal Peao cotiua [6]. Due to the similarities betwee H ad H, we call H the coarse Hawaiia earrig. 2. Fudametal Groups Let X be a topological space with basepoit x 0 X. A path i X is a cotiuous fuctio α : [0, 1] X. We say X is path-coected if every pair of poits x, y X ca be coected by a path p : [0, 1] X with p(0) = x ad p(1) = y. All spaces i this paper will be path-coected. We say a path p is a loop based at x 0 if α(0) = α(1). Let Ω(X, x 0 ) be the set of cotiuous fuctios p : [0, 1] X such that p(0) = p(1) = x 0. Let α : [0, 1] X be the reverse path of α defied as α (t) = α(1 t). If α ad β are paths i X satisfyig α(1) = β(0), let α β be the cocateatio defied piecewise as α(2t), 0 t 1 2 α β(t) = 1 β(2t 1), 2 t 1. More geerally, if α 1,..., α is a sequece of paths such that α i (1) = α i+1 (0) for i = 1,..., 1, let i=1 α i be the path defied as α i o the iterval [ i 1, ] i. Two loops α ad β based at x 0 are homotopic if there is a map H : [0, 1] [0, 1] X such that H(s, 0) = α(s), H(s, 1) = β(s) ad H(0, t) = H(1, t) = x 0 for all s, t [0, 1]. We write α β if α ad β are homotopic. Homotopy is a equivalece relatio o the set of loops Ω(X, x 0 ). The equivalece class [α] of a loop α is called the homotopy class of α. The set of homotopy classes π 1 (X, x 0 ) = Ω(X, x 0 )/ is called the fudametal group of X at x 0. It is a group whe it has multiplicatio [α] [β] = [α β] ad [α] 1 = [α ] is the iverse of [α] [11]. A space X is simply coected if X is path-coected ad π 1 (X, x 0 ) is isomorphic to the trivial group. Fially, a map f : X Y such that f (x 0 ) = y 0 iduces a well-defied homomorphism f : π 1 (X, x 0 ) π 1 (Y, y 0 ) give by f ([α]) = [ f α]. Fudametal groups are ofte studied usig maps called coverig maps. We refer to [11] ad [13] for the basic theory of algebraic topology. Defiitio 3. Let p : X X be a map. A ope set U X is evely covered by p if p 1 (U) X is the disjoit uio λ Λ V λ where V λ is ope i X ad p Vλ : V λ U is a homeomorphism for every λ Λ. A coverig map is a map p : X X such that every poit x X has a ope eighborhood which is evely covered by p. We call p a uiversal coverig map if X is simply coected. A importat property of coverig maps is that for every path α : [0, 1] X such that α(0) = x 0 ad poit y p 1 (x 0 ) there is a uique path α y : [0, 1] X (called a lift of α) such that p α y = p ad α y (0) = y.

3 A COUNTABLE SPACE WITH AN UNCOUNTABLE FUNDAMENTAL GROUP 3 Lemma 4. [11] A coverig map p : X X such that p(y) = x 0 iduces a ijective homomorphism p : π 1 ( X, y) π 1 (X, x 0 ). If α : [0, 1] X is a loop based at x 0, the [α] p (π 1 ( X, y)) if ad oly if α y (1) = y. A coverig map p : X X iduces a liftig correspodece map φ : π 1 (X, x 0 ) p 1 (x 0 ) from the fudametal group of X to the fiber over x 0. This is defied by the formula φ([α]) = α y (1). Lemma 5. [11] If p : X X is a coverig map, the the liftig correspodece φ : π 1 (X, x 0 ) p 1 (x 0 ) is surjective. If p is a uiversal coverig map, the p is bijective. Example 6. Let S 1 = {(x, y) x 2 + y 2 = 1} be the uit circle ad b 0 = (1, 0). The expoetial map ɛ : R S 1, ɛ(t) = (cos(2πt), si(2πt)) defied o the real lie is a coverig map such that ɛ 1 (0) = Z is the set of itegers. The liftig correspodece for this coverig map φ : π 1 (S 1, b 0 ) ɛ 1 (b 0 ) = Z is a isomorphism whe Z is the additive group of itegers. 3. Some basic fiite spaces A fiite space is a topological space X = {x 1, x 2,..., x } with fiitely may poits. Example 7. The coarse iterval is the three-poit space I = {0, 1/2, 1} with topology geerated by the basic sets the sets {0}, {1}, ad I. I other words, the topology of I is T I = {I, {0}, {1}, {0, 1}, }. Figure 1. The coarse iterval I. A basic ope set is illustrated here as a bouded regio whose iterior cotais the poits of the set. The coarse iterval is clearly T 0. It is also path-coected sice we ca defie a cotiuous surjectio p : [0, 1] I by 0, t [0, 1/2) p(t) = 1/2 t = 1/2 1, t (1/2, 1] ad the cotiuous image of a path-coected space is path-coected. A space X is cotractible if the idetity map id : X X is homotopic to a costat map X X. Every cotractible space is simply coected. Lemma 8. The coarse iterval I is cotractible. Proof. To show I is cotractible we defie a cotiuous map G : I [0, 1] I such that G(x, 0) = x for x I ad G(x, 1) = 1/2. The set C = ({0, 1} [1/2, 1]) ({1/2} [0, 1]) is closed i I [0, 1]. Defie G by 0, (s, t) {0} [0, 1/2) G(s, t) = 1/2, (s, t) C 1, (s, t) {1} [0, 1/2) This fuctio is well-defied ad cotiuous sice {0} ad {1} are ope i I.

4 4 JEREMY BRAZAS AND LUIS MATOS Corollary 9. I is simply coected. For = 0, 1, 2, 3, let b = ( cos ( ) ( )) π 2, cos π 2 S 1 be the poits of the uit circle o the coordiate axes, i.e. b 0 = (1, 0), b 1 = (0, 1), b 2 = ( 1, 0), ad b 3 = (0, 1). Example 10. The coarse circle is the four-poit set S = {b i i = 0, 1, 2, 3} with the topology geerated by the basic sets {b 0, b 1, b 2 }, {b 2, b 3, b 0 }, {b 0 }, ad {b 2 }. The etire topology of S may be writte as T S = {S, {b 0, b 1, b 2 }, {b 2, b 3, b 0 }, {b 0, b 2 }, {b 0 }, {b 2 }, }. Figure 2. The coarse circle S ad it s basic ope sets. Observe that the ope sets U 1 = {b 0, b 1, b 2 } ad U 2 = {b 2, b 3, b 0 } are homeomorphic to I whe they are give the subspace topology. Sice S is the uio of two path-coected subsets with o-empty itersectio, S is also path-coected. 4. The coarse lie as a coverig space The mai purpose of this sectio is to show that the fudametal group π 1 (S, b 0 ) of the coarse circle is isomorphic to the ifiite cyclic group Z, i.e. the additive group of itegers. Example 11. The coarse lie is the set L = { 4 R Z} with the topology geerated by the basis B cosistig of the sets A = { 2 } ad B = { 2, 2+1 4, +1 2 } for each Z. Eve though L is ot a fiite space, it is a coutable space with a T 0 but o-t 1 topology. Figure 3. The basic ope sets geeratig the topology of the coarse lie L. Lemma 12. L is simply coected. Proof. The set L = L [ 2, 2 ] is ope i L sice it is the uio of the basic sets B k = { k 2, 2k+1 4, k+1 2 }, k =,..., 1 with the subspace topology of L. We prove usig iductio that L is simply coected. Sice B k I for each k, B k is simply coected for each k. Observe that L 1 = B 1 B 0 where B 1 B 0 = {0} is simply coected sice it oly has oe poit. By the va Kampe Theorem [11], L 1 is simply coected. Now suppose L is simply coected. Sice L, B, ad

5 A COUNTABLE SPACE WITH AN UNCOUNTABLE FUNDAMENTAL GROUP 5 L B = { 2 } are all simply coected, L B is simply coected by the va Kampe theorem. Similarly, sice L B, B 1, ad (L B ) B k 1 = { 2 } are all simply coected, L +1 = B 1 L B is simply coected by the va Kampe theorem. This completes the proof by iductio. Sice L is the uio of the path-coected sets L all of which cotai 0, it follows that L is path-coected. Now suppose α : [0, 1] L is a path such that α(0) = α(1). Sice [0, 1] is compact, the image α([0, 1]) is compact. But {L 1} is a ope cover of L such that L L +1. Sice α must have image i a fiite subcover of {L 1}, we must have α([0, 1]) L for some. But L is simply coected, showig that α is homotopic to the costat loop at 0. This proves π 1 (L, 0) is the trivial group, i.e. L is simply coected. Just like the usual coverig map ɛ : R S 1 used to compute π 1 (S 1, b 0 ), we defie a similar coverig map i the coarse situatio. Example 13. Cosider the fuctio p : L S from the coarse lie to the coarse circle, which is the restrictio of the coverig map ɛ : R S 1. More directly, defie p ( ) 4 = b mod4. We check that each o-empty, ope set i S ca be writte as a uio of basic ope sets i L. Sice p 1 ({b 0 }) = Z = k Z A 2k p 1 ({b 2 }) = Z = k Z A 2k+1 p 1 (U 1 ) = k Z B 2k p 1 (U 2 ) = k Z B 2k+1 we ca coclude that p is cotiuous. Lemma 14. p : L S is a coverig map. Proof. We claim that the sets U 1, U 2 are evely covered by p. Notice that p 1 (U 1 ) = k Z B 2k is a disjoit uio where each B 2k is ope. Recall that both B 2k ad U 1 are homeomorphic to I; specifically p B2k : B 2k U 1 is a homeomorphism. Thus U 1 is evely covered. Similarly, p 1 (U 2 ) is the disjoit uio k Z B 2k+1 where each B 2k+1 is ope ad is mapped homeomorphically o to U 2 by p. Sice p : L S is a coverig map ad L is simply coected, p is a uiversal coverig map. The proof of the followig theorem is similar to the proof that the liftig correspodece for ɛ is a group isomorphism. We remark that eve though L is ot a topological group, for each Z, the shift map σ : L L, σ(t) = t + is a homeomorphism satisfyig p σ = p. Theorem 15. The liftig correspodece φ : π 1 (S, b 0 ) p 1 (b 0 ) = Z is a group isomorphism where Z has the usual additive group structure. Proof. Sice p : L S is a coverig map ad L is simply coected, φ is bijective by Lemma 5. Suppose α, β : [0, 1] S are loops based at b 0. Respectively, let α 0 : [0, 1] L ad β 0 : [0, 1] L be the uique lifts of α ad β startig at 0. Sice α 0 (1) p 1 (b 0 ) = Z, we have φ([α]) = α 0 (1) = for some iteger. Similarly, φ([β]) = β 0 (1) = m for some iteger m.

6 6 JEREMY BRAZAS AND LUIS MATOS Cosider the cocateated path γ = α 0 (σ β 0 ) : [0, 1] L from 0 to m +. Sice p σ = p, we have p γ = p ( α 0 (σ β 0 )) = (p α 0 ) (p σ β 0 ) = (p α 0 ) (p β 0 ) = α β which meas that γ is a lift of α β startig at 0. Sice lifts are uique, this meas γ = α β 0. It follows that φ([α][β]) = φ([α β]) = α β 0 (1) = γ(1) = m +. This proves φ is a group homomorphism. Both π 1 (S 1, b 0 ) ad π 1 (S, b 0 ) are isomorphic to the ifiite cyclic group Z. I fact, we ca defie maps which iduce the isomorphism betwee the two fudametal groups. Let f : R L be the map defied so that f (( 2 1 4, )) = 2 ad f ( ) = for each Z. Notice that p f is costat o each fiber ɛ 1 (x), x S 1. Therefore, there is a iduced map g : S 1 S such that g ɛ = p f. Propositio 16. The iduced homomorphism g : π 1 (S 1, b 0 ) π 1 (S, b 0 ) is a group isomorphism. Proof. Recall that ɛ 1 (b 0 ) = Z ad p 1 (b 0 ) = Z ad otice that the restrictio to the fibers f Z : Z Z is the idetity map. Let i : [0, 1] R be the iclusio ad ote f i : [0, 1] L is a path from 0 to 1. The group π 1 (S 1, b 0 ) is freely geerated by the homotopy class of α = ɛ i ad π 1 (S, b 0 ) is freely geerated by the homotopy class of p f i. Sice g ([ɛ i]) = [g ɛ i] = [p f i], g maps oe free geerator to the other ad it follows that g is a isomorphism. 5. The coarse Hawaiia earrig Let C = { (x, y) R 2 (x 1 )2 + y 2 = 1 2 } be the circle of radius 1 cetered at ( 1, 0). The Hawaiia earrig is the coutably ifiite uio H = 1 C of these circles over the positive itegers. We model this costructio by replacig the usual circle with the coarse circle studied i the previous sectios. Figure 4. The Hawaiia earrig H

7 A COUNTABLE SPACE WITH AN UNCOUNTABLE FUNDAMENTAL GROUP 7 Let w 0 = (0, 0), ad for itegers 1 defie x = ( 1, ) 1, y = ( 2, 0), ad z = ( 1, ) 1. Let D = {w 0, x, y, z } ad H = 1 D. Note that H is a coutable subset of H. Defiitio 17. The coarse Hawaiia earrig is the set H with the topology geerated by the basis cosistig of the followig sets for each 1: {x }, {z }, {x, y, z }, ad V = j D j {x 1} {z 1}. Notice that V = H\{y 1,..., y 1 } cotais all but fiitely may of the fiite circles D j. These sets form a eighborhood base at w 0 so that H is ot a Alexadrovdiscrete space i the sese of [1]. Figure 5. The uderlyig set of the coarse Hawaiia earrig H (left) ad the basic ope set V 8 (right). Propositio 18. H is a path-coected, locally path-coected, compact, T 0 space which is ot T 1. Proof. Notice that the set D H is homeomorphic to the coarse circle S whe equipped with the subspace topology of H. Sice D is path-coected ad w 0 D for each 1, it follows that H is path-coected. To see that H is locally path-coected, we check that every basic ope set is path-coected. Certaily, {x } ad {z } are path-coected. Sice {x, y, z } is homeomorphic to I whe it is give the subspace topology of H, this basic ope set is path-coected. Additioally, the subspace {w 0, x, y } H is homeomorphic to I ad is path-coected. Therefore, sice V is the uio j D 1{w 0, x, y } of sets all of which are path-coected ad cotai w 0, we ca coclude that V is path-coected. This proves H is locally path-coected. To see that H is compact let U be a ope cover of H. Sice the oly basic ope sets cotaiig w 0 are the sets V there must be a U 0 U such that w 0 V U 0 for some. For i = 1,..., 1, fid a set U i U such that y i U i. Now {U 0, U 1,..., U 1 } is a fiite subcover of U. This proves H is compact. To see that H is T 0, we pick two poits a, b H. If a = w 0 ad b = y, the a V +1 but b V +1. If a = w 0 ad b {x, z }, the b lies i the ope set {x, y, z } but a does ot. If a {x, z } ad a b, the {a} is ope ad does ot cotai b. This cocludes all the possible cases of distict pairs of poits i H provig that H is T 0.

8 8 JEREMY BRAZAS AND LUIS MATOS H is ot T 1 sice the every ope eighborhood V of w 0 cotais the ifiite set 1{w 0, x, z }. Sice D S, by Theorem 15, we have π 1 (D, w 0 ) Z for each 1. Recall that if A is a subspace of X, the a retractio is a map r : X A such that r A : A A is the idetity map. Propositio 19. For each 1, the fuctio r : H D which is the idetity o D ad collapses j D j to w 0 is a retractio. Proof. Sice D is a subspace of H it suffices to show r is cotiuous. We have r 1 ({x }) = {x }, r 1 ({z }) = {z }, r 1 ({x, y, z }) = {x, y, z }, ad r 1 ({w 0, x, y }) = {x } {y } V +1 j<{x j, y j, z j }. Sice the pullback of each basic ope set i D is the uio of basic ope sets i H, r is cotiuous. Corollary 20. H is ot semi-locally simply coected at w 0. Proof. Fix 1. We show that V cotais a loop α which is ot ull-homotopic i H. Let α : [0, 1] D be ay loop based at w 0 such that [α] is ot the idetity elemet of π 1 (D, w 0 ). Let i : D H be the iclusio map so that r i = id D is the idetity map. Sice π 1 is a fuctor, (r ) i = (r i) = id π1 (D,w 0 ) is the idetity homomorphism of π 1 (D, w 0 ). I particular, i α is a loop i H with image i D V such that (r ) ([i α]) = [α] is ot the idetity of π 1 (D, w 0 ). Sice homomorphisms preserve idetity elemets, [i α] caot be the idetity elemet of π 1 (H, w 0 ). Defiitio 21. The ifiite product of a sequece of groups G 1, G 2,... is deoted 1 G ad cosists of all ifiite sequeces (g 1, g 2,...) with g G for each 1. Group multiplicatio ad iversio are evaluated compoet-wise. If G = Z for each 1, the the group 1 Z cosistig of sequeces ( 1, 2,...) of itegers is called the Baer-Specker group. Ifiite products of groups have the useful property that if G is a fixed group ad f : G G is a sequece of homomorphisms, the there is a well-defied homomorphism f : G 1 G give by f (g) = ( f 1 (g), f 2 (g),...). Lemma 22. The ifiite product 1 π 1 (D, w 0 ) is ucoutable. Proof. If each G is o-trivial, the G cotais at least two elemets. Therefore the product 1 G is ucoutable sice the Cator set {0, 1} N = 1{0, 1} ca be ijected as a subset. I particular, the Baer-Specker group is ucoutable. Sice π 1 (D, w 0 ) Z for each 1, the ifiite product 1 π 1 (D, w 0 ) is isomorphic to the Baer-Specker group ad is therefore ucoutable. Let λ : [0, 1] D be the loop defied as w 0, t {0, 1} x, t (0, 1/2) λ (t) = y, t = 1/2 z, t (1/2, 1) This fuctio is cotiuous ad therefore a loop i D. I particular, our descriptio of the uiversal coverig of S i the previous sectio shows that the homotopy class [λ ] is a geerator of the cyclic group π 1 (D, b 0 ).

9 A COUNTABLE SPACE WITH AN UNCOUNTABLE FUNDAMENTAL GROUP 9 Defiitio 23. Suppose for each 1, we have a cotiuous loop α : [0, 1] H based at w 0 with image i D. The ifiite cocateatio of this sequece of loops is the [ loop α : [0, 1] H defied as follows: for each 1, the restrictio of α to 1, +1] is the path α ad α (1) = w 0. Lemma 24. α is a cotiuous loop such that [r α ] = [α ] for each 1. Proof. Sice each loop α is cotiuous ad each cocateatio α α +1 is cotiuous, it is eough to show that α is cotiuous at 1. Cosider a basic ope eighborhood ([ V of α (1) = w 0. Sice α i has image i V for each i, we have α 1, 1]) V. I particular, 1 ( 1, 1] f 1 (V ). This proves that α is cotiuous. Sice r α is defied as α o [ 1, ] +1 ad is costat at w0 o [ ] [ 0, 1 +1, 1] (ad the suitable arragemet whe = 1. Therefore r α is homotopic to α. Theorem 25. π 1 (H, w 0 ) is ucoutable. Proof. We have a sequece of homomorphisms (r ) : π 1 (H, w 0 ) π 1 (D, w 0 ). Together these iduce a homomorphism r : π 1 (H, w 0 ) 1 π 1 (D, w 0 ). give by r([α]) = ((r 1 ) ([α]), (r 2 ) ([α]),...) = ([r 1 α], [r 2 α],...). By Lemma 22, the ifiite product 1 π 1 (D, w 0 ) is ucoutable. We claim that r is oto. Suppose (g 1, g 2,...) 1 π 1 (D, w 0 ) where g π 1 (D, w 0 ). Sice g is a elemet of the ifiite cyclic group π 1 (D, w 0 ) geerated by [λ ], we may write g = [λ ] m for some iteger m Z. For each 1, defie a loop α as follows: m i=1 λ, if m > 0 α = costat at w 0, if m = 0 m i=1 λ, if m < 0 Notice that α is defied so that g = [λ ] m = [α ]. Let α : [0, 1] H be the loop based at w 0 which is the ifiite cocateatio as i Defiitio 23. By Lemma 24, we have [r α ] = [α ] = g for each 1. Therefore r([α ]) = (g 1, g 2,...). This proves that r is oto. Sice π 1 (H, b 0 ) surjects oto a ucoutable group, it must also be ucoutable. We coclude that there is a T 0 space with coutably may poits but which has a ucoutable fudametal group. Refereces [1] P.S. Alexadroff. Diskrete Räume. Mathematiceskii Sborik (N.S.) 2 (1937), [2] J. Barmak, Algebraic topology of fiite topological spaces ad applicatios, vol of Lecture Notes i Mathematics. Spriger, Heidelberg, [3] J.A. Barmak, E.G. Miia, Simple homotopy types ad fiite spaces. Adv. Math., 218 o. 1 (2008) [4] J.W. Cao, G.R. Coer, The combiatorial structure of the Hawaiia earrig group, Topology Appl. 106 (2000) [5] N. Ciaci, M. Ottia, Smallest homotopically trivial o-cotractible space. Preprit, arxiv: [6] K. Eda, Homotopy types of oe-dimesioal Peao cotiua, Fud. Math. 209 (2010) [7] Fiite topological spaces, Notes for REU (2003).

10 10 JEREMY BRAZAS AND LUIS MATOS [8] Fiite spaces ad simplicial complexes, Notes for REU (2003). [9] Fiite groups ad fiite spaces, Notes for REU (2003). [10] M.C. McCord, Sigular homology groups ad homotopy groups of fiite topological spaces. Duke Math. J. 33, 3 (1966), [11] J. Mukres, Topology, 2d Editio Pretice Hall, [12] W. Sierpiski, U thorme sur les cotius, Thoku Math. J. 13 (1918), [13] E. Spaier, Algebraic Topology, McGraw-Hill, [14] R. Stog, Fiite topological spaces, Tras. Amer. Math. Soc. 123 (1966),