A discrete form of Jordan curve theorem
Abstract
The paper includes purely combinatorial proof of a theorem that implies the following theorem, stated by Hugo Steinhaus in [6, p.35]: consider a chessboard (rectangular, not necessarily square) with some "mined" squares on. Assume that the king, while moving in accordance with the chess rules, cannot go across the chessboard from the left edge to the right one without meeting a mined square. Then the rook can go across the chessboard from the upper edge to the lower one moving exclusively on mined squares. All proofs of the Steinhaus theorem already published (see [5] and remarks on some proofs in [7, p.211]) are incomplete, except the hexagonal variant proved by Gale in [1].
Steinhaus theorem was thought in [6, p.269] as the lemma in the proof of the Brouwer fixed point theorem for the square (cf. Šaškin [5] and Gale [1]). It can also be used as the lemma for the mountain climbing theorem of Homma [2] (see Mioduszewski [3]). In this paper the Steinhaus theorem is used in the proof of a discrete analogue of the Jordan curve theorem (see Stout [8], where different proof is stated; cf. also Rosenfeld [4]).
References
2. T. Homma, A theorem on continuous functions, Kodai Math., Sem. Reports 1 (1952), 13-16.
3. J. Mioduszewski, The two climbers, Delta 101 (1982), 14-15 (in Polish).
4. A. Rosenfeld, Digital Topology, Amer. Math. Monthly 86 (1979), 621-630.
5. U.A. Šaškin, Fixed points, Nauka, Moscow 1989 (in Russian).
6. H. Steinhaus, Mathematical Snapshot, WSiP Warszawa 1989 (in Polish).
7. H. Steinhaus, Problems and Essays, Mir, Moscow 1974 (in Russian).
8. L.N. Stout, Two Discrete Forms of the Jordan Curve Theorem, Amer. Math. Monthly 95 (1988), 332-336.
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
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