Metric Space Completeness is not Preserved by Homeomorphism.
homeomorphism khaws cia li cas?
Ib homeomorphism, tseem hu ua kev hloov pauv tas mus li, yog qhov sib npaug sib npaug thiab ib-rau-ib tsab ntawv sib txuas ntawm cov ntsiab lus hauv ob daim duab geometric lossis qhov chaw topological uas txuas ntxiv mus rau ob qho tib si. Ib hom homeomorphism uas tseem khaws cia nrug yog hu ua isometry.
Puas yog homeomorphism khaws cia compactness?
3.3 Cov khoom ntawm qhov chaw sib cog lus
Peb tau sau ua ntej tias kev cog lus yog cov cuab yeej topological ntawm aspace, uas yog hais tias nws tau khaws cia los ntawm homeomorphism. Txawm ntau, nws tau khaws cia los ntawm ib qho kev ua haujlwm tas mus li.
Puas yog kev ua tiav yog cov cuab yeej topological?
Kev ua tiav tsis yog cov cuab yeej topological, piv txwv li ib qho tsis tuaj yeem xam seb qhov chaw metric ua tiav tsuas yog saib ntawm qhov chaw topological.
Vim li cas boundedness tsis yog topological khoom?
Rau qhov chaw ntsuas peb muaj qhov kev xav ntawm kev cia siab: uas yog qhov chaw metric yog bounded yog tias muaj qee tus lej M xws li d(x, y) ≤ M rau txhua x, y. Boundedness tsis yog cov cuab yeej topological. Piv txwv li, (0, 1) thiab (1, ∞) yog homeomorphic tab sis ib tug yog bounded thiab ib tug tsis yog. ∞ n=1 yog ib ntu ntawm cov ntsiab lus hauv X.
