Pseudo-Confluent Mappings and a Classification of Continua

Lelek, A.; Tymchatyn, E. D.

doi:10.4153/cjm-1975-136-4

Cited by 10 publications

(8 citation statements)

References 10 publications

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“…The proof of the following theorem is almost the same as the proof of Theorem 5.5 of [6]. A continuum X is acyclic if each map from X onto a circle is homotopic to a constant map.…”

Section: Theorem If F Is a Partially Confluent Map From A Hereditarimentioning

confidence: 58%

Maps which preserve graphs

Nall

1987

Proc. Amer. Math. Soc.

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ABSTRACT. In 1976 Eberhart, Fúgate, and Gordh proved that the weakly confluent image of a graph is a graph. A much weaker condition on the map is introduced called partial confluence, and it is shown that the image of a graph is a graph if and only if the map is partially confluent.In addition, it is shown that certain properties of one-dimensional continua are preserved by partially confluent maps, generalizing theorems of Cook and Lelek, Tymchatyn and Lelek, and Grace and Vought. Also, some continua in addition to graphs are shown to be the images of partially confluent maps only.

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“…The proof of the following theorem is almost the same as the proof of Theorem 5.5 of [6]. A continuum X is acyclic if each map from X onto a circle is homotopic to a constant map.…”

Section: Theorem If F Is a Partially Confluent Map From A Hereditarimentioning

confidence: 58%

Maps which preserve graphs

Nall

1987

Proc. Amer. Math. Soc.

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“…Theorem 8.1 generalizes earlier results about invariance of rational continua by proving that rational continua are preserved by jFf-pseudo confluent mappings. We also prove some invariance theorems of hereditarily (7-connected and hereditarily weakly d-connected spaces, and Theorem 10.6 generalizes Theorem 4.6 of [11].…”

Section: Introductionmentioning

confidence: 91%

“…Also, implications (2) and (5) are true provided that the mappings are perfect. It has been also proved that the implication (3) is invertible provided that X is a compact metric space and F is a continuum (see [13]), that (5) is invertible provided that F is a hereditarily locally connected continuum (see [3]), and that (6), (7), (11), (12), (16) and (17) are invertible provided that F is a locally connected complete metric space and X is a hereditarily normal space (see [11,Propositions 3…”

Section: Remark 32 It Is Easymentioning

confidence: 99%

“…Then the natural projection / of X onto F = X/R is a confluent mapping which is not /^-confluent (see [11,Example 3.7]).…”

Section: Proof Letmentioning

confidence: 99%

“…Let/ be a mapping such that/(0) = a 0 and/maps the intervals {(0, y): 0 ^ y ^ 1/3}, {(0, y): 1/3 ^ 3; ^ 2/3} and {(0,y): 2/3 ^ y ^ 1} homeomorphically onto the arcs A 0 \J A\, A\ VJ A 2 and ^4 2 U A 0 , respectively. Then /is i7-pseudo confluent since condition (iii) of Proposition 3.13 is satisfied but is not weakly confluent (see [11,Example 3.6] Conversely, let the condition be satisfied and let U be an open neighborhood of a component C of/ _1 (;yo). If yo d Int /(£/)> then there exist points y n £ Y\f(U) (n = 1, 2, .…”

Section: Proof Letmentioning

confidence: 99%

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