Definition df-nlly 23450

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 23448	. 2 class 𝑛-Locally 𝐴
3		vj	. . . . . . . . 9 setvar 𝑗
4	3	cv 1546	. . . . . . . 8 class 𝑗
5		vu	. . . . . . . . 9 setvar 𝑢
6	5	cv 1546	. . . . . . . 8 class 𝑢
7		crest 17374	. . . . . . . 8 class ↾t
8	4, 6, 7	co 7356	. . . . . . 7 class (𝑗 ↾t 𝑢)
9	8, 1	wcel 2119	. . . . . 6 wff (𝑗 ↾t 𝑢) ∈ 𝐴
10		vy	. . . . . . . . . 10 setvar 𝑦
11	10	cv 1546	. . . . . . . . 9 class 𝑦
12	11	csn 4555	. . . . . . . 8 class {𝑦}
13		cnei 23080	. . . . . . . . 9 class nei
14	4, 13	cfv 6485	. . . . . . . 8 class (nei‘𝑗)
15	12, 14	cfv 6485	. . . . . . 7 class ((nei‘𝑗)‘{𝑦})
16		vx	. . . . . . . . 9 setvar 𝑥
17	16	cv 1546	. . . . . . . 8 class 𝑥
18	17	cpw 4529	. . . . . . 7 class 𝒫 𝑥
19	15, 18	cin 3882	. . . . . 6 class (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)
20	9, 5, 19	wrex 3063	. . . . 5 wff ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴
21	20, 10, 17	wral 3053	. . . 4 wff ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴
22	21, 16, 4	wral 3053	. . 3 wff ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴
23		ctop 22876	. . 3 class Top
24	22, 3, 23	crab 3391	. 2 class {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴}
25	2, 24	wceq 1547	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  23452  nllyeq  23454