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

Definition df-nlly 21490
Description: Define a space that is n-locally 𝐴, where 𝐴 is a topological property like "compact", "connected", or "path-connected". A topological space is n-locally 𝐴 if every neighborhood of a point contains a sub-neighborhood that is 𝐴 in the subspace topology.

The terminology "n-locally", where 'n' stands for "neighborhood", is not standard, although this is sometimes called "weakly locally 𝐴". The reason for the distinction is that some notions only make sense for arbitrary neighborhoods (such as "locally compact", which is actually 𝑛-Locally Comp in our terminology - open compact sets are not very useful), while others such as "locally connected" are strictly weaker notions if the neighborhoods are not required to be open. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion
Ref Expression
df-nlly 𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴}
Distinct variable group:   𝑢,𝑗,𝑥,𝑦,𝐴

Detailed syntax breakdown of Definition df-nlly
StepHypRef Expression
1 cA . . 3 class 𝐴
21cnlly 21488 . 2 class 𝑛-Locally 𝐴
3 vj . . . . . . . . 9 setvar 𝑗
43cv 1629 . . . . . . . 8 class 𝑗
5 vu . . . . . . . . 9 setvar 𝑢
65cv 1629 . . . . . . . 8 class 𝑢
7 crest 16288 . . . . . . . 8 class t
84, 6, 7co 6792 . . . . . . 7 class (𝑗t 𝑢)
98, 1wcel 2144 . . . . . 6 wff (𝑗t 𝑢) ∈ 𝐴
10 vy . . . . . . . . . 10 setvar 𝑦
1110cv 1629 . . . . . . . . 9 class 𝑦
1211csn 4314 . . . . . . . 8 class {𝑦}
13 cnei 21121 . . . . . . . . 9 class nei
144, 13cfv 6031 . . . . . . . 8 class (nei‘𝑗)
1512, 14cfv 6031 . . . . . . 7 class ((nei‘𝑗)‘{𝑦})
16 vx . . . . . . . . 9 setvar 𝑥
1716cv 1629 . . . . . . . 8 class 𝑥
1817cpw 4295 . . . . . . 7 class 𝒫 𝑥
1915, 18cin 3720 . . . . . 6 class (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)
209, 5, 19wrex 3061 . . . . 5 wff 𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴
2120, 10, 17wral 3060 . . . 4 wff 𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴
2221, 16, 4wral 3060 . . 3 wff 𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴
23 ctop 20917 . . 3 class Top
2422, 3, 23crab 3064 . 2 class {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴}
252, 24wceq 1630 1 wff 𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴}
Colors of variables: wff setvar class
This definition is referenced by:  isnlly  21492  nllyeq  21494
  Copyright terms: Public domain W3C validator