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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 23944 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (topGen‘ran (ball‘𝐷))) |
3 | blbas 23936 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) ∈ TopBases) | |
4 | tgtopon 22474 | . . . 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 23926 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋) | |
7 | 6 | fveq2d 6896 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (TopOn‘∪ ran (ball‘𝐷)) = (TopOn‘𝑋)) |
8 | 5, 7 | eleqtrd 2836 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (topGen‘ran (ball‘𝐷)) ∈ (TopOn‘𝑋)) |
9 | 2, 8 | eqeltrd 2834 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∪ cuni 4909 ran crn 5678 ‘cfv 6544 topGenctg 17383 ∞Metcxmet 20929 ballcbl 20931 MetOpencmopn 20934 TopOnctopon 22412 TopBasesctb 22448 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-n0 12473 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-topgen 17389 df-psmet 20936 df-xmet 20937 df-bl 20939 df-mopn 20940 df-top 22396 df-topon 22413 df-bases 22449 |
This theorem is referenced by: mopntop 23946 mopnuni 23947 mopnm 23950 mopnss 23952 isxms2 23954 methaus 24029 prdsxmslem2 24038 metcnp3 24049 metcn 24052 metcnpi3 24055 txmetcn 24057 cnfldms 24292 cnfldtopn 24298 metdseq0 24370 metdscn2 24373 iitopon 24395 lebnumlem2 24478 lmmbr 24775 cfilfcls 24791 cmetcaulem 24805 iscmet3lem2 24809 lmle 24818 nglmle 24819 caublcls 24826 metcnp4 24827 metcn4 24828 metsscmetcld 24832 cmetss 24833 relcmpcmet 24835 bcth2 24847 vmcn 29952 dipcn 29973 blocni 30058 ipasslem7 30089 ubthlem1 30123 ubthlem2 30124 minvecolem4b 30131 minvecolem4 30133 axhcompl-zf 30251 hlimadd 30446 hlim0 30488 occllem 30556 hmopidmchi 31404 fmcncfil 32911 ismtyhmeolem 36672 heiborlem9 36687 bfplem2 36691 |
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