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Unicoherent Space

المؤلف:  Charatonik, J. J. and Prajs, J. R.

المصدر:  "On Local Connectedness of Absolute Retracts." Pacific J. Math. 201

الجزء والصفحة:  ...

30-7-2021

2108

Unicoherent Space

Let  be a connected topological space. Then  is unicoherent provided that for any closed connected subsets  and  of , if , then  is connected.

An interval, say [0,1], is unicoherent, but a circle, say {e^(itheta):theta in [0,2pi]} subset= C" src="https://mathworld.wolfram.com/images/equations/UnicoherentSpace/Inline8.gif" style="height:21px; width:154px" />, is not unicoherent. An interesting example of a unicoherent space is a ray winding down on a circle. Specifically, let , where {(1+1/(1+theta))e^(itheta):0<=theta<infty} subset= C" src="https://mathworld.wolfram.com/images/equations/UnicoherentSpace/Inline10.gif" style="height:23px; width:198px" />. Then the space , illustrated above, is unicoherent.

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REFERENCES:

Charatonik, J. J. and Prajs, J. R. "On Local Connectedness of Absolute Retracts." Pacific J. Math. 201, 83-88, 2001.

Mackowiak, T. "Retracts of Hereditarily Unicoherent Continua." Bull. Acad. Polon. Sci. Ser. Sci. Math. 28, 177-183, 1980.