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- W2023873740 abstract "It is proved that the arbitrary product of arcwise connected spaces is arcwise connected. Introduction. By an arc we mean a Hausdorff continuum with at most 2 noncut points, called the end points of the arc. A space S is said to be arcwise connected if whenever x, y E S, then x and y are the end points of some arc in S. It is well known (see, for instance, [4, Theorems 28.8 and 28.13]) that a nondegenerate metric continuum A is an arc if and only if A is homeomorphic to [0, 1]. Since a metrizable product of arcs is a compact, connected and locally connected metric space, it follows [4, Theorem 31.2] that a metrizable product of arcs is arcwise connected. However, examples have been constructed by S. Mardesic [2] and [3] and J. L. Cornette and B. Lehman [1] of locally connected Hausdorff continua which are not arcwise connected. Thus the above argument will not suffice for a nonmetrizable product of arcs, even if each factor space is metrizable. In this paper we show that the arbitrary product of arcwise connected spaces is arcwise connected. LEMMA Let {X,: a E W} be a collection of nondegenerate arcs, and let X denote the product space of this collection. If the end points of Xa are a. and ba, then there is an arc in Xfromf to g ii-heref is that pointfor which f(c)=a. and g is that point for which g(oc)=ba. PROOF. Let ? be a well-ordering of s', and let denote the first element of a?, and oc+ 1 the successor of a. in a. For each c E a., define the edge A. of X and pointsfa and g. of X as follows: Aa = {h e X:h(/3) = b#,5 c., = apl > a; =ap , > a. Received by the editors May 4, 1973. AMS (MOS) subject classifications (1970). Primary 54B10, 54F05, 54F20." @default.
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- W2023873740 date "1974-01-01" @default.
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- W2023873740 title "Products of arcwise connected spaces" @default.
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- W2023873740 doi "https://doi.org/10.1090/s0002-9939-1974-0346762-8" @default.
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