() @ Appl. Gen. Topol. 16, no. 1(2015), 15-17doi:10.4995/agt.2015.1826 c© AGT, UPV, 2015 Contractibility of the digital n-space Sayaka Hamada Department of Mathematics, Yatsushiro Campus, National Institute of Technology, Kumamoto College, 866-8501 Japan. (hamada@kumamoto-nct.ac.jp) Abstract The aim of this paper is to prove a known fact that the digital line is contractible. Hence we have that the digital space (Zn, κn) is also contractible where (Zn, κn) is n products of the digital line (Z, κ). This is a fundamental property of homotopy theory. 2010 MSC: 14F35; 54B10. Keywords: Khalimsky topology; digital n-space; contractible; homotopy. 1. Prelimarilies We consider an important property of homotopy theory for the digital n- space. The digital line (Z, κ) is the set of the integers Z equipped with the topology κ having {{2m − 1, 2m, 2m + 1} : m ∈ Z} as a subbase. For x ∈ Z, we set U(x) := { {2m − 1, 2m, 2m + 1} if x = 2m, {2m + 1} if x = 2m + 1. Then {U(x)} is a fundamental neighborhood system at x. Then it it obvious that {2m : m ∈ Z} is closed and nowhere dense in Z, {2m+ 1 : m ∈ Z} is open and dense in Z. U(x) is the minimal open set containing x for any x ∈ Z. (See [1],[2],[4],[5]). The digital line (Z, κ) was introduced by E. Khalimsky in the late 1960’s and it was made use of studying topological properties of digital images. (See [3], [6]). Received 4 November 2013 – Accepted 23 September 2014 http://dx.doi.org/10.4995/agt.2015.1826 S. Hamada The digital n-space (Zn, κn) is the topological product of n copies of the digital line (Z, κ). To investigate the digital n-space is very interesting for the application pos- sibility. Here we focus the contractibility of one. 2. Contractibility of the digital line and digital n-space A space X is called contractible provided that there exists a homotopy H : X×I → X such that HX×{0} is the identity and HX×{1} is a constant function. The digital line is contractible as pointed out in Remark 4.11 of [7]. We shall show by direct computation. Theorem 2.1. The digital line is contractible. Proof. Defining H : Z × I → Z by H{0}×I ≡ 0 and for any n ∈ Z\{0}, if n is an odd number, H(n, t) :=                    n if 0 ≤ t < 2−|n|, n − 1(if n > 0), n + 1(if n < 0) if 2−|n| ≤ t ≤ 2−(|n|−1), n − 2(if n > 0), n + 2(if n < 0) if 2−(|n|−1) < t < 2−(|n|−2), . . . 1(if n > 0), −1(if n < 0) if 2−2 < t < 2−1, 0 if 2−1 ≤ t, if n is an even number, H(n, t) :=                    n if 0 ≤ t ≤ 2−|n|, n − 1(if n > 0), n + 1(if n < 0) if 2−|n| < t < 2−(|n|−1), n − 2(if n > 0), n + 2(if n < 0) if 2−(|n|−1) ≤ t ≤ 2−(|n|−2), . . . 1(if n > 0), −1(if n < 0) if 2−2 < t < 2−1, 0 if 2−1 ≤ t, then we see that HZ×{0} = idZ and HZ×{1} ≡ 0. Since H is continuous, idZ and the constant map (≡ 0) is homotopic. Therefore we have (Z, κ) is contractible. � Since a contractible finite product of contractible spaces is contractible, we have the following. Corollary 2.2. The digital n-space is contractible. c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 16 Contractibility of the digital n-space References [1] M. Fujimoto, S. Takigawa, J. Dontchev, H. Maki and T. Noiri, The topological structures and groups of digital n-spaces, Kochi J. Math. 1(2006), 31–55. [2] S. Hamada and T. Hayashi, Fuzzy topological structures of low dimensional digital spaces, Journal of Fuzzy Mathematics 20, no. 1 (2012), 15–23. [3] E. D. Khalimsky, On topologies of generalized segments, Soviet Math. Doklady 10(1969) 1508–1511. [4] E. Khalimsky, R. Kopperman and P. R. Meyer, Computer graphics and connected topolo- gies on finite ordered sets, Topology Appl, 36(1990), 1–17. [5] T. Y. Kong, R. Kopperman and P. R. Meyer, A topological approach to digital topology, Am. Math. Monthly 98(1991), 901–917. [6] E. H. Kronheimer, The topology of digital images, Topology Appl. 46(1992), 279–303. [7] G. Raptis, Homotopy theory of posets, Homology, Homotopy and Applications 12, no. 2 (2010), 211–230. c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 17