Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-t0 Structured version   Visualization version   GIF version

Definition df-t0 21111
 Description: Define T0 or Kolmogorov spaces. A T0 space satisfies a kind of "topological extensionality" principle (compare ax-ext 2601): any two points which are members of the same open sets are equal, or in contraposition, for any two distinct points there is an open set which contains one point but not the other. This differs from T1 spaces (see ist1-2 21145) in that in a T1 space you can choose which point will be in the open set and which outside; in a T0 space you only know that one of the two points is in the set. (Contributed by Jeff Hankins, 1-Feb-2010.)
Assertion
Ref Expression
df-t0 Kol2 = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)}
Distinct variable group:   𝑗,𝑜,𝑥,𝑦

Detailed syntax breakdown of Definition df-t0
StepHypRef Expression
1 ct0 21104 . 2 class Kol2
2 vx . . . . . . . . 9 setvar 𝑥
3 vo . . . . . . . . 9 setvar 𝑜
42, 3wel 1990 . . . . . . . 8 wff 𝑥𝑜
5 vy . . . . . . . . 9 setvar 𝑦
65, 3wel 1990 . . . . . . . 8 wff 𝑦𝑜
74, 6wb 196 . . . . . . 7 wff (𝑥𝑜𝑦𝑜)
8 vj . . . . . . . 8 setvar 𝑗
98cv 1481 . . . . . . 7 class 𝑗
107, 3, 9wral 2911 . . . . . 6 wff 𝑜𝑗 (𝑥𝑜𝑦𝑜)
112, 5weq 1873 . . . . . 6 wff 𝑥 = 𝑦
1210, 11wi 4 . . . . 5 wff (∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)
139cuni 4434 . . . . 5 class 𝑗
1412, 5, 13wral 2911 . . . 4 wff 𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)
1514, 2, 13wral 2911 . . 3 wff 𝑥 𝑗𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)
16 ctop 20692 . . 3 class Top
1715, 8, 16crab 2915 . 2 class {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)}
181, 17wceq 1482 1 wff Kol2 = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)}
 Colors of variables: wff setvar class This definition is referenced by:  ist0  21118
 Copyright terms: Public domain W3C validator