Definition df-nlly 23371

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 23369	. 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 17343	. . . . . . . 8 class ↾t
8	4, 6, 7	co 7353	. . . . . . 7 class (𝑗 ↾t 𝑢)
9	8, 1	wcel 2109	. . . . . 6 wff (𝑗 ↾t 𝑢) ∈ 𝐴
10		vy	. . . . . . . . . 10 setvar 𝑦
11	10	cv 1539	. . . . . . . . 9 class 𝑦
12	11	csn 4579	. . . . . . . 8 class {𝑦}
13		cnei 23001	. . . . . . . . 9 class nei
14	4, 13	cfv 6486	. . . . . . . 8 class (nei‘𝑗)
15	12, 14	cfv 6486	. . . . . . 7 class ((nei‘𝑗)‘{𝑦})
16		vx	. . . . . . . . 9 setvar 𝑥
17	16	cv 1539	. . . . . . . 8 class 𝑥
18	17	cpw 4553	. . . . . . 7 class 𝒫 𝑥
19	15, 18	cin 3904	. . . . . 6 class (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)
20	9, 5, 19	wrex 3053	. . . . 5 wff ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴
21	20, 10, 17	wral 3044	. . . 4 wff ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴
22	21, 16, 4	wral 3044	. . 3 wff ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴
23		ctop 22797	. . 3 class Top
24	22, 3, 23	crab 3396	. 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  23373  nllyeq  23375