TYPICAL FAMILIES AND $T$-COMPACTNESS IN PROXIMITY SPACES

Raghad Almohammed; Dunia M.k. Al-Ftlawy; Nadiha Abed Habeeb; Luay Al. Swidi

TYPICAL FAMILIES AND $T$-COMPACTNESS IN PROXIMITY SPACES

Abstract

In this work we introduce a family of subsets of a proximity space, called the typical family, defined as the complement of a cluster family, and we study its main properties and its relation to the cluster concept. We show explicitly that the typical family is, up to the language of proximity spaces, the ideal that is dual to the associated cluster in the classical sense of Choquet's grill ideal duality, and we compare it in detail with ideals, filters, grills, and bornologies, establishing a necessary and sufficient condition under which it is a bornology. Building on this family we define a generalized compactness notion, called $T$-compactness, relate it to the classical notion of compactness with respect to an ideal, and establish preservation results for finite unions, closed subsets, and via a pushforward construction introduced here images under proximity isomorphisms. We illustrate the framework with a fully worked finite numeric example. We close with a discussion of the broader implications of these results for the relationship between cluster theory and generalized compactness.

Keywords

Thanks

The authors would like to extend their gratitude to their respective institutions for their support.

References

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Details

Primary Language

English

Subjects

Topology

Journal Section

Research Article

Authors

Raghad Almohammed ^* This is me
0000-0003-0559-0436
Iraq

Dunia M.k. Al-Ftlawy This is me
0009-0007-3600-1389
Iraq

Nadiha Abed Habeeb This is me
0009-0008-3552-744X
Iraq

Luay Al. Swidi This is me
0000-0001-5641-1616
Iraq

Publication Date

October 6, 2026

Submission Date

July 15, 2025

Acceptance Date

October 30, 2025

Published in Issue

Year 2026 Volume: 16 Number: 10

IZ

https://izlik.org/JA98FT24TP

Cite

RIS / Bibtex

APA

Almohammed, R., Al-Ftlawy, D. M., Habeeb, N. A., & Al. Swidi, L. (2026). TYPICAL FAMILIES AND $T$-COMPACTNESS IN PROXIMITY SPACES. TWMS Journal of Applied and Engineering Mathematics, 16(10), 1286-1296. https://izlik.org/JA98FT24TP

AMA

1.Almohammed R, Al-Ftlawy DM, Habeeb NA, Al. Swidi L. TYPICAL FAMILIES AND $T$-COMPACTNESS IN PROXIMITY SPACES. JAEM. 2026;16(10):1286-1296. https://izlik.org/JA98FT24TP

Chicago

Almohammed, Raghad, Dunia M.k. Al-Ftlawy, Nadiha Abed Habeeb, and Luay Al. Swidi. 2026. “TYPICAL FAMILIES AND $T$-COMPACTNESS IN PROXIMITY SPACES”. TWMS Journal of Applied and Engineering Mathematics 16 (10): 1286-96. https://izlik.org/JA98FT24TP.

EndNote

Almohammed R, Al-Ftlawy DM, Habeeb NA, Al. Swidi L (October 1, 2026) TYPICAL FAMILIES AND $T$-COMPACTNESS IN PROXIMITY SPACES. TWMS Journal of Applied and Engineering Mathematics 16 10 1286–1296.

IEEE

[1]R. Almohammed, D. M. Al-Ftlawy, N. A. Habeeb, and L. Al. Swidi, “TYPICAL FAMILIES AND $T$-COMPACTNESS IN PROXIMITY SPACES”, JAEM, vol. 16, no. 10, pp. 1286–1296, Oct. 2026, [Online]. Available: https://izlik.org/JA98FT24TP

ISNAD

Almohammed, Raghad - Al-Ftlawy, Dunia M.k. - Habeeb, Nadiha Abed - Al. Swidi, Luay. “TYPICAL FAMILIES AND $T$-COMPACTNESS IN PROXIMITY SPACES”. TWMS Journal of Applied and Engineering Mathematics 16/10 (October 1, 2026): 1286-1296. https://izlik.org/JA98FT24TP.

JAMA

1.Almohammed R, Al-Ftlawy DM, Habeeb NA, Al. Swidi L. TYPICAL FAMILIES AND $T$-COMPACTNESS IN PROXIMITY SPACES. JAEM. 2026;16:1286–1296.

MLA

Almohammed, Raghad, et al. “TYPICAL FAMILIES AND $T$-COMPACTNESS IN PROXIMITY SPACES”. TWMS Journal of Applied and Engineering Mathematics, vol. 16, no. 10, Oct. 2026, pp. 1286-9, https://izlik.org/JA98FT24TP.

Vancouver

1.Raghad Almohammed, Dunia M.k. Al-Ftlawy, Nadiha Abed Habeeb, Luay Al. Swidi. TYPICAL FAMILIES AND $T$-COMPACTNESS IN PROXIMITY SPACES. JAEM [Internet]. 2026 Oct. 1;16(10):1286-9. Available from: https://izlik.org/JA98FT24TP