![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > df-nlly | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
df-nlly | ⊢ 𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | 1 | cnlly 22070 | . 2 class 𝑛-Locally 𝐴 |
3 | vj | . . . . . . . . 9 setvar 𝑗 | |
4 | 3 | cv 1537 | . . . . . . . 8 class 𝑗 |
5 | vu | . . . . . . . . 9 setvar 𝑢 | |
6 | 5 | cv 1537 | . . . . . . . 8 class 𝑢 |
7 | crest 16686 | . . . . . . . 8 class ↾t | |
8 | 4, 6, 7 | co 7135 | . . . . . . 7 class (𝑗 ↾t 𝑢) |
9 | 8, 1 | wcel 2111 | . . . . . 6 wff (𝑗 ↾t 𝑢) ∈ 𝐴 |
10 | vy | . . . . . . . . . 10 setvar 𝑦 | |
11 | 10 | cv 1537 | . . . . . . . . 9 class 𝑦 |
12 | 11 | csn 4525 | . . . . . . . 8 class {𝑦} |
13 | cnei 21702 | . . . . . . . . 9 class nei | |
14 | 4, 13 | cfv 6324 | . . . . . . . 8 class (nei‘𝑗) |
15 | 12, 14 | cfv 6324 | . . . . . . 7 class ((nei‘𝑗)‘{𝑦}) |
16 | vx | . . . . . . . . 9 setvar 𝑥 | |
17 | 16 | cv 1537 | . . . . . . . 8 class 𝑥 |
18 | 17 | cpw 4497 | . . . . . . 7 class 𝒫 𝑥 |
19 | 15, 18 | cin 3880 | . . . . . 6 class (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) |
20 | 9, 5, 19 | wrex 3107 | . . . . 5 wff ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴 |
21 | 20, 10, 17 | wral 3106 | . . . 4 wff ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴 |
22 | 21, 16, 4 | wral 3106 | . . 3 wff ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴 |
23 | ctop 21498 | . . 3 class Top | |
24 | 22, 3, 23 | crab 3110 | . 2 class {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴} |
25 | 2, 24 | wceq 1538 | 1 wff 𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴} |
Colors of variables: wff setvar class |
This definition is referenced by: isnlly 22074 nllyeq 22076 |
Copyright terms: Public domain | W3C validator |