About w c s -covers and w c s * -networks on the Vietoris hyperspace ℱ ( X )

Luong Quoc Tuyen; Ong V. Tuyen; Phan D. Tuan; Nguzen X. Truc

Commentationes Mathematicae Universitatis Carolinae (2023)

  • Volume: 64, Issue: 4, page 519-527
  • ISSN: 0010-2628

Abstract

top
We study some generalized metric properties on the hyperspace ℱ ( X ) of finite subsets of a space X endowed with the Vietoris topology. We prove that X has a point-star network consisting of (countable) w c s -covers if and only if so does ℱ ( X ) . Moreover, X has a sequence of w c s -covers with property ( P ) which is a point-star network if and only if so does ℱ ( X ) , where ( P ) is one of the following properties: point-finite, point-countable, compact-finite, compact-countable, locally finite, locally countable. On the other hand, X has a w c s * -network with property σ - ( P ) if and only if so does ℱ ( X ) . By these results, we obtain some results related to the images of metric spaces and separable metric spaces under some kinds of continuous mappings on the Vietoris hyperspace ℱ ( X ) .

How to cite

top

Tuyen, Luong Quoc, et al. "About $wcs$-covers and $wcs^*$-networks on the Vietoris hyperspace $\mathcal {F}(X)$." Commentationes Mathematicae Universitatis Carolinae 64.4 (2023): 519-527. <http://eudml.org/doc/299324>.

@article{Tuyen2023,
abstract = {We study some generalized metric properties on the hyperspace $\mathcal \{F\}(X)$ of finite subsets of a space $X$ endowed with the Vietoris topology. We prove that $X$ has a point-star network consisting of (countable) $wcs$-covers if and only if so does $\mathcal \{F\}(X)$. Moreover, $X$ has a sequence of $wcs$-covers with property $(P)$ which is a point-star network if and only if so does $\mathcal \{F\}(X)$, where $(P)$ is one of the following properties: point-finite, point-countable, compact-finite, compact-countable, locally finite, locally countable. On the other hand, $X$ has a $wcs^*$-network with property $\sigma $-$(P)$ if and only if so does $\mathcal \{F\}(X)$. By these results, we obtain some results related to the images of metric spaces and separable metric spaces under some kinds of continuous mappings on the Vietoris hyperspace $\mathcal \{F\}(X)$.},
author = {Tuyen, Luong Quoc, Tuyen, Ong V., Tuan, Phan D., Truc, Nguzen X.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {hyperspace; generalized metric property; $wcs$-cover; $wcs^*$-network},
language = {eng},
number = {4},
pages = {519-527},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {About $wcs$-covers and $wcs^*$-networks on the Vietoris hyperspace $\mathcal \{F\}(X)$},
url = {http://eudml.org/doc/299324},
volume = {64},
year = {2023},
}

TY - JOUR
AU - Tuyen, Luong Quoc
AU - Tuyen, Ong V.
AU - Tuan, Phan D.
AU - Truc, Nguzen X.
TI - About $wcs$-covers and $wcs^*$-networks on the Vietoris hyperspace $\mathcal {F}(X)$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2023
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 64
IS - 4
SP - 519
EP - 527
AB - We study some generalized metric properties on the hyperspace $\mathcal {F}(X)$ of finite subsets of a space $X$ endowed with the Vietoris topology. We prove that $X$ has a point-star network consisting of (countable) $wcs$-covers if and only if so does $\mathcal {F}(X)$. Moreover, $X$ has a sequence of $wcs$-covers with property $(P)$ which is a point-star network if and only if so does $\mathcal {F}(X)$, where $(P)$ is one of the following properties: point-finite, point-countable, compact-finite, compact-countable, locally finite, locally countable. On the other hand, $X$ has a $wcs^*$-network with property $\sigma $-$(P)$ if and only if so does $\mathcal {F}(X)$. By these results, we obtain some results related to the images of metric spaces and separable metric spaces under some kinds of continuous mappings on the Vietoris hyperspace $\mathcal {F}(X)$.
LA - eng
KW - hyperspace; generalized metric property; $wcs$-cover; $wcs^*$-network
UR - http://eudml.org/doc/299324
ER -

References

top
  1. An T. V., Tuyen L. Q., On π -images of separable metric spaces and a problem of Shou Lin, Matematički Vesnik 64 (250) (2012), 297–302. MR2965962
  2. Gao Z. M., ℵ -space is invariant under perfect mappings, Questions Answer Gen. Topology 5 (1987), no. 2, 271–279. MR0917885
  3. Ge Y., On pseudo-sequence coverings, π -images of metric spaces, Mat. Vesnik 57 (2005), no. 3–4, 113–120. MR2194600
  4. Ge Y., Gu J. S., On π -images of separable metric spaces, Sci. Ser. A Math. Sci. (N.S.) 10 (2004), 65–71. MR2127483
  5. Good C., Macías S., Symmetric products of generalized metric spaces, Topology Appl. 206 (2016), 93–114. MR3494434
  6. Guthrie J. A., 10.1016/0016-660X(71)90116-4, General Topology and Appl. 1 (1971), no. 2, 105–110. MR0288726DOI10.1016/0016-660X(71)90116-4
  7. Li J. J., Images of a Locally Separable Metric Space and Their Associated Results, Doctoral Thesis, Shandong University, Jinan, 2000 (in Chinese). 
  8. Li Z., 10.1155/IJMMS.2005.1101, Int. J. Math. Math. Sci. 7 (2005), no. 7, 1101–1107. MR2170507DOI10.1155/IJMMS.2005.1101
  9. Lin F., Shen R., Liu C., Generalized metric properties on hyperspaces with the Vietoris topology, Rocky Mountain J. Math. 51 (2021), no. 5, 1761–1779. MR4382997
  10. Lin S., Point-countable Covers and Sequence-Covering Mappings, China Science Press, Beijing, 2015 (in Chinese). MR1939779
  11. Lin S., Tanaka Y., 10.1016/0166-8641(94)90101-5, Topology Appl. 59 (1994), no. 1, 79–86. Zbl0817.54025MR1293119DOI10.1016/0166-8641(94)90101-5
  12. Liu C., Lin F., 10.1007/s40840-022-01349-2, Bull. Malays. Math. Sci. Soc. 45 (2022), no. 5, 1955–1974. MR4489545DOI10.1007/s40840-022-01349-2
  13. Liu C., Lin F., Hyperspaces with a countable character of closed subset, Topology Appl. 328 (2023), Paper No. 108461, 14 pages. MR4553037
  14. Liu C., Lin F., The quasi-metrizability of hyperspaces, Topology Appl. 338 (2023), Paper No. 108665, 11 pages. MR4629790
  15. Mou L., Li P., Lin S., Regular G δ -diagonals and hyperspaces, Topology Appl. 301 (2021), Paper No. 107530, 9 pages. MR4312980
  16. Peng L.-X., Sun Y., 10.1016/j.topol.2017.09.036, Topology Appl. 231 (2017), 411–429. MR3712980DOI10.1016/j.topol.2017.09.036
  17. Tanaka Y., Ge Y., Around quotient compact images of metric spaces, and symmetric spaces, Houston J. Math. 32 (2006), no. 1, 99–117. MR2202355
  18. Tang Z., Lin S., Lin F., 10.1016/j.topol.2017.11.004, Topology Appl. 234 (2018), 26–45. MR3739454DOI10.1016/j.topol.2017.11.004
  19. Tuyen L. Q., Tuyen O. V., On the n -fold symmetric product of a space with a σ - ( P ) -property c n -network ( c k -network), Comment. Math. Univ. Carolinae 61 (2020), no. 2, 257–263. MR4143708
  20. Tuyen L. Q., Tuyen O. V., A note on the hyperspace of finite subsets, Fasc. Math. 65 (2021), 67–73. MR4478558
  21. Tuyen L. Q., Tuyen O. V., Kočinac L. D. R., 10.15672/hujms.1203236, Hacet. J. Math. Stat. 53 (2024), no. 2, 356–366. MR4741786DOI10.15672/hujms.1203236
  22. Yan P., On strong sequence-covering compact mappings, Northeast. Math. J. 14 (1998), no. 3, 341–344. Zbl0927.54030MR1685267

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.