A fixed-point theorem

Wallace, A. D.

doi:10.1090/s0002-9904-1945-08367-0

Cited by 36 publications

(19 citation statements)

References 6 publications

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“…of a result of Wallace [3] and Theorem 2 is equivalent to a result of Wang [5]. Then K and L are t-invariant continua and X = K^JL.…”

Section: Theorem 1 Below Is a Restatementmentioning

confidence: 81%

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Topological transformation groups with a fixed end point

Gray¹

1967

Proc. Amer. Math. Soc.

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“…of a result of Wallace [3] and Theorem 2 is equivalent to a result of Wang [5]. Then K and L are t-invariant continua and X = K^JL.…”

Section: Theorem 1 Below Is a Restatementmentioning

confidence: 81%

“…Hausdorff space and T is a topological group which leaves an end point e of X fixed. In [3], Wallace proved that if T is cyclic and X is a locally connected continuum, T has a fixed point other than e. Then in [4], Wallace asked the following question: If X is a peano continuum and T is compact or abelian, then does T have a fixed point other than e?…”

mentioning

confidence: 99%

Topological transformation groups with a fixed end point

Gray¹

1967

Proc. Amer. Math. Soc.

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“…[12]. Il indique également que le résultat est signalé, sous une forme différente, par G. Birkhoff dans la deuxième édition de [4] (1948).…”

Section: R(x)/xex};unclassified

Éléments maximaux des préordres partiels sur les ensembles compacts

Geistdoerfer-Florenzano¹

1978

RAIRO-Oper. Res.

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“…This means that Kx C H; order Kx in a standard fashion as follows: e is the first element of Kx (assuming e e K~x); if p, y eKx, p =£ e, y =£ e, then p ^ y iff p = y or p separates e from y in X. ^ is a total order on Kx-By virtue of [8], Kx has a largest element w; w is a cut point of X, and w is evidently fixed under K. Further, every point of Cw separates e from Cz. We note that Cw is compact and proceed as above to show that Cw contains a fixed point of C. But this means that w is fixed under C. Certainly w =fc e, so that C and K have a fixed point, other than e, in common.…”

Section: If X Is Irreducibly T-invariant Then X Contains No Cut Pointmentioning

confidence: 99%

A note on group invariant continua

Gray

1968

J. Aust. Math. Soc.

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Let (X, T, n) be a topological transformation group, where X is a Hausdorff continuum. We will say that X is irreducibly T-invariant if no proper subcontinuum of X is ^-invariant. Wallace, [6], has shown that if T is abelian and X is irreducibly ^-invariant, then X has no cut point; he then asked if this statement remains true if "abelian" is replaced by "compact". In this paper we answer this question in the affirmative, and prove a related result when T satisfies a recursive property.A general problem due to Wallace is as follows: Assume T leaves an endpoint of X fixed. Under what conditions on X and T does T have another fixed point? This problem has been investigated by Wallace [8], Wang, [5], Chu, [1], and Gray, [3,4]. In Theorem 2, we show that if X is locally connected, and T is generated by a compact subgroup and a connected subgroup, then T has another fixed point.Departing from [7] slightly, we call a subcontinuum C of X a universal subcontinuum (USC) if given a subcontinuum D of X, D n C is a continuum. The intersection of arbitrarily many USC is again a USC. If X-x = U u V, where U and V are non-empty separated subsets of X (hereafter referred to as a "separation of X-x"), then U u {x} is a USC. The property of being a USC is topological. The proofs of these statements are to be found in [7].The terminology pertaining to transformation groups is taken from [2]. THEOREM 1. Let (X, T, n) be a topological transformation group where X is a Hausdorff continuum and one of the following conditions is satisfied:(i) T is compact, (ii) X is locally connected and T is pointwise regularly almost periodic. If X is irreducibly T-invariant, then X contains no cut point.PROOF. Assume that X is irreducibly T-invariant. We make the following observations:1. If x e X, no proper USC of X contains the orbit Tx of x.For otherwise the intersection, D, of all USC which contains Tx is a proper subcontinuum of X containing Tx. We easily verify that that D is T-invariant, and hence X is not irreducibly T-invariant. 310

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A fixed-point theorem

Cited by 36 publications

References 6 publications

Topological transformation groups with a fixed end point

Topological transformation groups with a fixed end point

Éléments maximaux des préordres partiels sur les ensembles compacts

A note on group invariant continua

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