# Two types of remainders of topological groups

Commentationes Mathematicae Universitatis Carolinae (2008)

- Volume: 49, Issue: 1, page 119-126
- ISSN: 0010-2628

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topArhangel'skii, Aleksander V.. "Two types of remainders of topological groups." Commentationes Mathematicae Universitatis Carolinae 49.1 (2008): 119-126. <http://eudml.org/doc/250448>.

@article{Arhangelskii2008,

abstract = {We prove a Dichotomy Theorem: for each Hausdorff compactification $bG$ of an arbitrary topological group $G$, the remainder $bG\setminus G$ is either pseudocompact or Lindelöf. It follows that if a remainder of a topological group is paracompact or Dieudonne complete, then the remainder is Lindelöf, and the group is a paracompact $p$-space. This answers a question in A.V. Arhangel’skii, Some connections between properties of topological groups and of their remainders, Moscow Univ. Math. Bull. 54:3 (1999), 1–6. It is shown that every Tychonoff space can be embedded as a closed subspace in a pseudocompact remainder of some topological group. We also establish some other results and present some examples and questions.},

author = {Arhangel'skii, Aleksander V.},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {remainder; compactification; topological group; $p$-space; Lindelöf $p$-space; metrizability; countable type; Lindelöf space; pseudocompact space; $\pi $-base; compactification; remainder; compactification; topological group; -space},

language = {eng},

number = {1},

pages = {119-126},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {Two types of remainders of topological groups},

url = {http://eudml.org/doc/250448},

volume = {49},

year = {2008},

}

TY - JOUR

AU - Arhangel'skii, Aleksander V.

TI - Two types of remainders of topological groups

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2008

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 49

IS - 1

SP - 119

EP - 126

AB - We prove a Dichotomy Theorem: for each Hausdorff compactification $bG$ of an arbitrary topological group $G$, the remainder $bG\setminus G$ is either pseudocompact or Lindelöf. It follows that if a remainder of a topological group is paracompact or Dieudonne complete, then the remainder is Lindelöf, and the group is a paracompact $p$-space. This answers a question in A.V. Arhangel’skii, Some connections between properties of topological groups and of their remainders, Moscow Univ. Math. Bull. 54:3 (1999), 1–6. It is shown that every Tychonoff space can be embedded as a closed subspace in a pseudocompact remainder of some topological group. We also establish some other results and present some examples and questions.

LA - eng

KW - remainder; compactification; topological group; $p$-space; Lindelöf $p$-space; metrizability; countable type; Lindelöf space; pseudocompact space; $\pi $-base; compactification; remainder; compactification; topological group; -space

UR - http://eudml.org/doc/250448

ER -

## References

top- Arhangel'skii A.V., On a class of spaces containing all metric and all locally compact spaces, Mat. Sb. 67 (109) (1965), 55-88; English translation: Amer. Math. Soc. Transl. 92 (1970), 1-39. (1965) MR0190889
- Arhangel'skii A.V., 10.1070/RM1981v036n03ABEH004249, Russian Math. Surveys 36 (3) (1981), 151-174. (1981) MR0622722DOI10.1070/RM1981v036n03ABEH004249
- Arhangel'skii A.V., Some connections between properties of topological groups and of their remainders, Moscow Univ. Math. Bull. 54:3 (1999), 1-6. (1999) MR1711899
- Arhangel'skii A.V., Topological invariants in algebraic environment, in: Recent Progress in General Topology 2, eds. M. Hušek, Jan van Mill, North-Holland, Amsterdam, 2002, pp.1-57. Zbl1030.54026MR1969992
- Arhangel'skii A.V., 10.1016/j.topol.2004.10.015, Topology Appl. 150 (2005), 79-90. (2005) Zbl1075.54012MR2133669DOI10.1016/j.topol.2004.10.015
- Arhangel'skii A.V., 10.1016/j.topol.2006.10.008, Topology Appl. 154 (2007), 1084-1088. (2007) Zbl1144.54001MR2298623DOI10.1016/j.topol.2006.10.008
- Engelking R., General Topology, PWN, Warszawa, 1977. Zbl0684.54001MR0500780
- Filippov V.V., On perfect images of paracompact $p$-spaces, Soviet Math. Dokl. 176 (1967), 533-536. (1967) MR0222853
- Henriksen M., Isbell J.R., 10.1215/S0012-7094-58-02509-2, Duke Math. J. 25 (1958), 83-106. (1958) Zbl0081.38604MR0096196DOI10.1215/S0012-7094-58-02509-2
- Tkachenko M.G., The Suslin property in free topological groups over compact spaces (Russian), Mat. Zametki 34 (1983), 601-607; English translation: Math. Notes 34 (1983), 790-793. (1983) MR0722229
- Roelke W., Dierolf S., Uniform Structures on Topological Groups and their Quotients, McGraw-Hill, New York, 1981.

## Citations in EuDML Documents

top- Aleksander V. Arhangel'skii, The Baire property in remainders of topological groups and other results
- Aleksander V. Arhangel'skii, J. van Mill, Nonnormality of remainders of some topological groups
- Liang-Xue Peng, Yu-Feng He, A note on topological groups and their remainders
- Aleksander V. Arhangel'skii, Miroslav Hušek, Closed embeddings into complements of $\Sigma $-products
- D. Basile, Angelo Bella, About remainders in compactifications of homogeneous spaces
- Aleksander V. Arhangel'skii, A generalization of Čech-complete spaces and Lindelöf $\Sigma $-spaces

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