Wp/lus/Hmunramzirna

Möbius-a puanthem

Hmunramzirna hi chhiarkawp péng lian tak pakhat a ni a, péng tam tak lian pui puia ṭhen lehchhawn a ni. Hliam/tihkeh/tihthlèr awm lova taksa herhsawihin a vawnṭhat nihphungte chhuina a ni. Entirnan: Rubber phêkah hian pentuiin rin bial la, rubber chu pawt thlër lovin pawt sàwi tâ la, a pentui chuan bial kaihsawih a la ziak reng a. Chuvangin i bial ziah khan bialna chu vawngṭha lo mah sé, inzawmna a la vawngṭha: a bial kha a rubber i pawh thlèr hma chuan a inzawm reng tho vang. I bial siamin nihphung a vawnṭhat chu inzawmna a ni, bialna erawh a vawngṭha lo thung. Chuvangin hemi chungchang bîkah chuan inzawmna kha vawnṭhat-nihphung kan ti.

Hmunramzirna chuan hetiang a entirna hmanga sawifiah theih leh entirna hmanga sawifiah theih lëm loh nihphung tam tak a zir a ni.

Lo chhuah dàn

Hmunramzirna hi leitehna chhiarkawpa zawhna ṭhenkhat chhàn tumna aṭanga lo inṭan chho a ni a; Leonhard Euler-a'n kum 1736-a K¨nigsberg Lei Pasarihte a zirbingna thuziak kha hmunramzirna thuziak mumal hmasa bera ngaih a ni.

Hmunramzirna vuah chhan

Hmunramzirna hi sap ṭawng chuan topology tih a ni a, chu thumal chu zerman thumal Topologie tih aṭanga thlâk rem a ni a, chu zerman thumal chu Grik thumal τόπος (topos), chu chu hmun tihna, leh λόγος (logos), chu chu zirna tihna, aṭanga lâk lehchhawn a ni. He zerman thumal Topologie tih hi kum 1847-a Johann Benedict Listing-a lehkhabu Vorstudien zur Topologie tiha ziah hmasak ber a ni a, amaherawhchu JB Listing-a hian ziaka a chhuah hma kum 10 vêl zetah a thukhawchang sawinaah a lo hmang tawh a ni.

A bulbal

Tunlai hmunramzirna hi khawnkhawmzirna nèn ṭhenhran hleih theih lohvin a inkai bet tlat a, hmunramzirna hi khawnkhawmzirna thukhawchang hmang lo chuan zir theih loh a ni.

A chhiarkawp

Chanchin kimchang : Hmunram

$X$ hi khawnkhâwm lo ni ta se, $\tau$ hi $X$ khawnpéng chhungkua lo ni ta bawk se. (Chhungkua, khawnpéng tih te hi a phêk hrangah kan hrilhfiah ang). $\tau$ hian a hnuaia nihphung pathumte khu a neih chuan $X$ hmunramthlá tiin kan sawi ang:

1. khawnkhawm ruak $\emptyset$ leh $X$ te hi $\tau$-ah a awm ve ve.
2. $\tau$ chhungkhung engzât pawh (chhiarsen loh tiamin) suihkhâwmin $\tau$ chhungkhung bawk a chhuak.
3. $\tau$ chhungkhung bichin intawhnain $\tau$ chhungkhung bawk a siam chhuak.

Hetiang a chunga nihphung pathumte khi $\tau$-vin a neih chuan, kan sawi lâwk angin, $\tau$ hi hmunramthla kan ti a, kawpchawi ($X,\tau$) hi Hmunram kan ti. A hmunramthla $\tau$ hi hriat sâ emaw, bituk sâ emaw a nih chuan, thil awlsam zâwk nan "$X$ hi Hmunram a ni" kan ti mai bawk ṭhin.

Entirna

1. Khawnkhawm $X$ hi $X=\lbrace 1,2,3,4\rbrace$ lo ni ta se, $\tau :=\lbrace \lbrace \rbrace ,\lbrace 1,2,3,4\rbrace \rbrace$ hi hmunramthla a ni. Hei hi a të leh mawlmang thei ang ber a nih avangin hmunram holam tiin kan ko.
2. Khawnkhawm $X$ hi $X=\lbrace 1,2,3,4\rbrace$ lo ni leh ta se, $\tau :=\lbrace \lbrace \rbrace ,\lbrace 2\rbrace ,\lbrace 1,2\rbrace ,\lbrace 2,3\rbrace ,\lbrace 1,2,3\rbrace ,\lbrace 1,2,3,4\rbrace \rbrace$ lo ni ve leh thung ta se. $\tau$ hi hmunramthla a ni. Chutiang zëlin.

Thulâkna

• Munkres, James R., Topology, 2nd Edition, Prentice Hall, 1975
• von Querenburg, Boto, Mengentheoretische Topologie, 3. Auflage, Springer-Verlag, 2001