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Definition df-nlly 23491
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 subneighborhood 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 23489 . 2 class 𝑛-Locally 𝐴
3 vj . . . . . . . . 9 setvar 𝑗
43cv 1536 . . . . . . . 8 class 𝑗
5 vu . . . . . . . . 9 setvar 𝑢
65cv 1536 . . . . . . . 8 class 𝑢
7 crest 17467 . . . . . . . 8 class t
84, 6, 7co 7431 . . . . . . 7 class (𝑗t 𝑢)
98, 1wcel 2106 . . . . . 6 wff (𝑗t 𝑢) ∈ 𝐴
10 vy . . . . . . . . . 10 setvar 𝑦
1110cv 1536 . . . . . . . . 9 class 𝑦
1211csn 4631 . . . . . . . 8 class {𝑦}
13 cnei 23121 . . . . . . . . 9 class nei
144, 13cfv 6563 . . . . . . . 8 class (nei‘𝑗)
1512, 14cfv 6563 . . . . . . 7 class ((nei‘𝑗)‘{𝑦})
16 vx . . . . . . . . 9 setvar 𝑥
1716cv 1536 . . . . . . . 8 class 𝑥
1817cpw 4605 . . . . . . 7 class 𝒫 𝑥
1915, 18cin 3962 . . . . . 6 class (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)
209, 5, 19wrex 3068 . . . . 5 wff 𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴
2120, 10, 17wral 3059 . . . 4 wff 𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴
2221, 16, 4wral 3059 . . 3 wff 𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴
23 ctop 22915 . . 3 class Top
2422, 3, 23crab 3433 . 2 class {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴}
252, 24wceq 1537 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  23493  nllyeq  23495
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