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Definition df-nlly 21210
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 21208 . 2 class 𝑛-Locally 𝐴
3 vj . . . . . . . . 9 setvar 𝑗
43cv 1479 . . . . . . . 8 class 𝑗
5 vu . . . . . . . . 9 setvar 𝑢
65cv 1479 . . . . . . . 8 class 𝑢
7 crest 16021 . . . . . . . 8 class t
84, 6, 7co 6615 . . . . . . 7 class (𝑗t 𝑢)
98, 1wcel 1987 . . . . . 6 wff (𝑗t 𝑢) ∈ 𝐴
10 vy . . . . . . . . . 10 setvar 𝑦
1110cv 1479 . . . . . . . . 9 class 𝑦
1211csn 4155 . . . . . . . 8 class {𝑦}
13 cnei 20841 . . . . . . . . 9 class nei
144, 13cfv 5857 . . . . . . . 8 class (nei‘𝑗)
1512, 14cfv 5857 . . . . . . 7 class ((nei‘𝑗)‘{𝑦})
16 vx . . . . . . . . 9 setvar 𝑥
1716cv 1479 . . . . . . . 8 class 𝑥
1817cpw 4136 . . . . . . 7 class 𝒫 𝑥
1915, 18cin 3559 . . . . . 6 class (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)
209, 5, 19wrex 2909 . . . . 5 wff 𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴
2120, 10, 17wral 2908 . . . 4 wff 𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴
2221, 16, 4wral 2908 . . 3 wff 𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴
23 ctop 20638 . . 3 class Top
2422, 3, 23crab 2912 . 2 class {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴}
252, 24wceq 1480 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  21212  nllyeq  21214
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