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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 sub-neighborhood 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 21639 | . 2 class 𝑛-Locally 𝐴 |
3 | vj | . . . . . . . . 9 setvar 𝑗 | |
4 | 3 | cv 1655 | . . . . . . . 8 class 𝑗 |
5 | vu | . . . . . . . . 9 setvar 𝑢 | |
6 | 5 | cv 1655 | . . . . . . . 8 class 𝑢 |
7 | crest 16434 | . . . . . . . 8 class ↾t | |
8 | 4, 6, 7 | co 6905 | . . . . . . 7 class (𝑗 ↾t 𝑢) |
9 | 8, 1 | wcel 2164 | . . . . . 6 wff (𝑗 ↾t 𝑢) ∈ 𝐴 |
10 | vy | . . . . . . . . . 10 setvar 𝑦 | |
11 | 10 | cv 1655 | . . . . . . . . 9 class 𝑦 |
12 | 11 | csn 4397 | . . . . . . . 8 class {𝑦} |
13 | cnei 21272 | . . . . . . . . 9 class nei | |
14 | 4, 13 | cfv 6123 | . . . . . . . 8 class (nei‘𝑗) |
15 | 12, 14 | cfv 6123 | . . . . . . 7 class ((nei‘𝑗)‘{𝑦}) |
16 | vx | . . . . . . . . 9 setvar 𝑥 | |
17 | 16 | cv 1655 | . . . . . . . 8 class 𝑥 |
18 | 17 | cpw 4378 | . . . . . . 7 class 𝒫 𝑥 |
19 | 15, 18 | cin 3797 | . . . . . 6 class (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) |
20 | 9, 5, 19 | wrex 3118 | . . . . 5 wff ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴 |
21 | 20, 10, 17 | wral 3117 | . . . 4 wff ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴 |
22 | 21, 16, 4 | wral 3117 | . . 3 wff ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴 |
23 | ctop 21068 | . . 3 class Top | |
24 | 22, 3, 23 | crab 3121 | . 2 class {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴} |
25 | 2, 24 | wceq 1656 | 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 21643 nllyeq 21645 |
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