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Mirrors > Home > MPE Home > Th. List > mopntopon | Structured version Visualization version GIF version |
Description: The set of open sets of a metric space 𝑋 is a topology on 𝑋. Remark in [Kreyszig] p. 19. This theorem connects the two concepts and makes available the theorems for topologies for use with metric spaces. (Contributed by Mario Carneiro, 24-Aug-2015.) |
Ref | Expression |
---|---|
mopnval.1 | ⊢ 𝐽 = (MetOpen‘𝐷) |
Ref | Expression |
---|---|
mopntopon | ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mopnval.1 | . . 3 ⊢ 𝐽 = (MetOpen‘𝐷) | |
2 | 1 | mopnval 23045 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (topGen‘ran (ball‘𝐷))) |
3 | blbas 23037 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) ∈ TopBases) | |
4 | tgtopon 21576 | . . . 4 ⊢ (ran (ball‘𝐷) ∈ TopBases → (topGen‘ran (ball‘𝐷)) ∈ (TopOn‘∪ ran (ball‘𝐷))) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (topGen‘ran (ball‘𝐷)) ∈ (TopOn‘∪ ran (ball‘𝐷))) |
6 | unirnbl 23027 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋) | |
7 | 6 | fveq2d 6649 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (TopOn‘∪ ran (ball‘𝐷)) = (TopOn‘𝑋)) |
8 | 5, 7 | eleqtrd 2892 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (topGen‘ran (ball‘𝐷)) ∈ (TopOn‘𝑋)) |
9 | 2, 8 | eqeltrd 2890 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∪ cuni 4800 ran crn 5520 ‘cfv 6324 topGenctg 16703 ∞Metcxmet 20076 ballcbl 20078 MetOpencmopn 20081 TopOnctopon 21515 TopBasesctb 21550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-bl 20086 df-mopn 20087 df-top 21499 df-topon 21516 df-bases 21551 |
This theorem is referenced by: mopntop 23047 mopnuni 23048 mopnm 23051 mopnss 23053 isxms2 23055 methaus 23127 prdsxmslem2 23136 metcnp3 23147 metcn 23150 metcnpi3 23153 txmetcn 23155 cnfldms 23381 cnfldtopn 23387 metdseq0 23459 metdscn2 23462 iitopon 23484 lebnumlem2 23567 lmmbr 23862 cfilfcls 23878 cmetcaulem 23892 iscmet3lem2 23896 lmle 23905 nglmle 23906 caublcls 23913 metcnp4 23914 metcn4 23915 metsscmetcld 23919 cmetss 23920 relcmpcmet 23922 bcth2 23934 vmcn 28482 dipcn 28503 blocni 28588 ipasslem7 28619 ubthlem1 28653 ubthlem2 28654 minvecolem4b 28661 minvecolem4 28663 axhcompl-zf 28781 hlimadd 28976 hlim0 29018 occllem 29086 hmopidmchi 29934 fmcncfil 31284 ismtyhmeolem 35242 heiborlem9 35257 bfplem2 35261 |
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