# Ends and quasicomponents

Open Mathematics (2010)

• Volume: 8, Issue: 6, page 1009-1015
• ISSN: 2391-5455

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## Abstract

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Let X be a connected locally compact metric space. It is known that if X is locally connected, then the space of ends (Freudenthal ends), EX, can be represented as the inverse limit of the set (space) S(X C) of components of X C, i.e., as the inverse limit of the inverse system $EX=\underset{←}{lim}\left(S\left(X\setminus C\right)\right),inclusions,CcompactinX\right)$ . In this paper, the above result is significantly improved. It is shown that for a space which is not locally connected, we can replace the space of components by the space of quasicomponents Q(X C) of X C. The following result is proved: if X is a connected locally compact metric space, then $EX=\underset{←}{lim}\left(Q\left(X\setminus C\right)\right),inclusions,CcompactinX\right)$ .

## How to cite

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Nikita Shekutkovski, and Gorgi Markoski. "Ends and quasicomponents." Open Mathematics 8.6 (2010): 1009-1015. <http://eudml.org/doc/269720>.

@article{NikitaShekutkovski2010,
abstract = {Let X be a connected locally compact metric space. It is known that if X is locally connected, then the space of ends (Freudenthal ends), EX, can be represented as the inverse limit of the set (space) S(X C) of components of X C, i.e., as the inverse limit of the inverse system $EX = \mathop \{\lim \}\limits \_ \leftarrow (S(X\backslash C)),inclusions,CcompactinX)$ . In this paper, the above result is significantly improved. It is shown that for a space which is not locally connected, we can replace the space of components by the space of quasicomponents Q(X C) of X C. The following result is proved: if X is a connected locally compact metric space, then $EX = \mathop \{\lim \}\limits \_ \leftarrow (Q(X\backslash C)),inclusions,CcompactinX)$ .},
author = {Nikita Shekutkovski, Gorgi Markoski},
journal = {Open Mathematics},
keywords = {Ends; Quasicomponent; Inverse limit; Covering by disjoint open sets; space of ends; inverse limit; quasi-component},
language = {eng},
number = {6},
pages = {1009-1015},
title = {Ends and quasicomponents},
url = {http://eudml.org/doc/269720},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Nikita Shekutkovski
AU - Gorgi Markoski
TI - Ends and quasicomponents
JO - Open Mathematics
PY - 2010
VL - 8
IS - 6
SP - 1009
EP - 1015
AB - Let X be a connected locally compact metric space. It is known that if X is locally connected, then the space of ends (Freudenthal ends), EX, can be represented as the inverse limit of the set (space) S(X C) of components of X C, i.e., as the inverse limit of the inverse system $EX = \mathop {\lim }\limits _ \leftarrow (S(X\backslash C)),inclusions,CcompactinX)$ . In this paper, the above result is significantly improved. It is shown that for a space which is not locally connected, we can replace the space of components by the space of quasicomponents Q(X C) of X C. The following result is proved: if X is a connected locally compact metric space, then $EX = \mathop {\lim }\limits _ \leftarrow (Q(X\backslash C)),inclusions,CcompactinX)$ .
LA - eng
KW - Ends; Quasicomponent; Inverse limit; Covering by disjoint open sets; space of ends; inverse limit; quasi-component
UR - http://eudml.org/doc/269720
ER -

## References

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1. [1] Ball B.J., Proper shape retracts, Fund. Math., 89(2), 1975, 177–189 Zbl0321.55029
2. [2] Ball B.J., Quasicompactifications and shape theory, Pacific J. Math., 84(2), 1979, 251–259 Zbl0394.54020
3. [3] Dugundji J., Topology, Series in Advanced Mathematics, Allyn and Bacon, Boston, 1966
4. [4] Engelking R., General Topology, 2nd ed., Sigma Ser. Pure Math., 6, Heldermann, Berlin, 1989
6. [6] Kuratowski K., Topology, Vol. 2, Mir, Moscow, 1969 (in Russian)
7. [7] Michael E., Cuts, Acta Math., 111(1), 1964, 1–36 http://dx.doi.org/10.1007/BF02391006
8. [8] Shekutkovski N., Vasilevska V., Equivalence of different definitions of space of ends, God. Zb. Inst. Mat. Prir.-Mat. Fak. Univ. Kiril Metodij Skopje, 39, 2001, 7–13 Zbl1054.54024

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