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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 23587 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (topGen‘ran (ball‘𝐷))) |
3 | blbas 23579 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) ∈ TopBases) | |
4 | tgtopon 22117 | . . . 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 23569 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋) | |
7 | 6 | fveq2d 6773 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (TopOn‘∪ ran (ball‘𝐷)) = (TopOn‘𝑋)) |
8 | 5, 7 | eleqtrd 2843 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (topGen‘ran (ball‘𝐷)) ∈ (TopOn‘𝑋)) |
9 | 2, 8 | eqeltrd 2841 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ∪ cuni 4845 ran crn 5590 ‘cfv 6431 topGenctg 17144 ∞Metcxmet 20578 ballcbl 20580 MetOpencmopn 20583 TopOnctopon 22055 TopBasesctb 22091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 ax-pre-sup 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-er 8479 df-map 8598 df-en 8715 df-dom 8716 df-sdom 8717 df-sup 9177 df-inf 9178 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-n0 12232 df-z 12318 df-uz 12580 df-q 12686 df-rp 12728 df-xneg 12845 df-xadd 12846 df-xmul 12847 df-topgen 17150 df-psmet 20585 df-xmet 20586 df-bl 20588 df-mopn 20589 df-top 22039 df-topon 22056 df-bases 22092 |
This theorem is referenced by: mopntop 23589 mopnuni 23590 mopnm 23593 mopnss 23595 isxms2 23597 methaus 23672 prdsxmslem2 23681 metcnp3 23692 metcn 23695 metcnpi3 23698 txmetcn 23700 cnfldms 23935 cnfldtopn 23941 metdseq0 24013 metdscn2 24016 iitopon 24038 lebnumlem2 24121 lmmbr 24418 cfilfcls 24434 cmetcaulem 24448 iscmet3lem2 24452 lmle 24461 nglmle 24462 caublcls 24469 metcnp4 24470 metcn4 24471 metsscmetcld 24475 cmetss 24476 relcmpcmet 24478 bcth2 24490 vmcn 29055 dipcn 29076 blocni 29161 ipasslem7 29192 ubthlem1 29226 ubthlem2 29227 minvecolem4b 29234 minvecolem4 29236 axhcompl-zf 29354 hlimadd 29549 hlim0 29591 occllem 29659 hmopidmchi 30507 fmcncfil 31875 ismtyhmeolem 35956 heiborlem9 35971 bfplem2 35975 |
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