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Mirrors > Home > MPE Home > Th. List > t1top | Structured version Visualization version GIF version |
Description: A T1 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.) |
Ref | Expression |
---|---|
t1top | ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | ist1 22806 | . 2 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽{𝑥} ∈ (Clsd‘𝐽))) |
3 | 2 | simplbi 499 | 1 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∀wral 3062 {csn 4626 ∪ cuni 4906 ‘cfv 6539 Topctop 22376 Clsdccld 22501 Frect1 22792 |
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-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-br 5147 df-iota 6491 df-fv 6547 df-t1 22799 |
This theorem is referenced by: t1t0 22833 lpcls 22849 perfcls 22850 restt1 22852 t1sep2 22854 sst1 22859 t1connperf 22921 t1hmph 23276 qtopt1 32752 onint1 35271 |
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