Definition df-nlly 23291

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 ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴}

Distinct variable group:   𝑢,𝑗,𝑥,𝑦,𝐴

Detailed syntax breakdown of Definition df-nlly

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2	1	cnlly 23289	. 2 class 𝑛-Locally 𝐴
3		vj	. . . . . . . . 9 setvar 𝑗
4	3	cv 1539	. . . . . . . 8 class 𝑗
5		vu	. . . . . . . . 9 setvar 𝑢
6	5	cv 1539	. . . . . . . 8 class 𝑢
7		crest 17373	. . . . . . . 8 class ↾t
8	4, 6, 7	co 7412	. . . . . . 7 class (𝑗 ↾t 𝑢)
9	8, 1	wcel 2105	. . . . . 6 wff (𝑗 ↾t 𝑢) ∈ 𝐴
10		vy	. . . . . . . . . 10 setvar 𝑦
11	10	cv 1539	. . . . . . . . 9 class 𝑦
12	11	csn 4628	. . . . . . . 8 class {𝑦}
13		cnei 22921	. . . . . . . . 9 class nei
14	4, 13	cfv 6543	. . . . . . . 8 class (nei‘𝑗)
15	12, 14	cfv 6543	. . . . . . 7 class ((nei‘𝑗)‘{𝑦})
16		vx	. . . . . . . . 9 setvar 𝑥
17	16	cv 1539	. . . . . . . 8 class 𝑥
18	17	cpw 4602	. . . . . . 7 class 𝒫 𝑥
19	15, 18	cin 3947	. . . . . 6 class (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)
20	9, 5, 19	wrex 3069	. . . . 5 wff ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴
21	20, 10, 17	wral 3060	. . . 4 wff ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴
22	21, 16, 4	wral 3060	. . 3 wff ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴
23		ctop 22715	. . 3 class Top
24	22, 3, 23	crab 3431	. 2 class {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴}
25	2, 24	wceq 1540	1 wff 𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴}

This definition is referenced by:  isnlly  23293  nllyeq  23295