Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > df-nlly | Structured version Visualization version Unicode version |
Description: Define a space that is
n-locally , where is a topological
property like "compact", "connected", or
"path-connected". A
topological space is n-locally if every neighborhood of a point
contains a sub-neighborhood that is in the subspace topology.
The terminology "n-locally", where 'n' stands for "neighborhood", is not standard, although this is sometimes called "weakly locally ". The reason for the distinction is that some notions only make sense for arbitrary neighborhoods (such as "locally compact", which is actually 𝑛Locally 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 | 𝑛Locally ↾t |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 | |
2 | 1 | cnlly 21268 | . 2 𝑛Locally |
3 | vj | . . . . . . . . 9 | |
4 | 3 | cv 1482 | . . . . . . . 8 |
5 | vu | . . . . . . . . 9 | |
6 | 5 | cv 1482 | . . . . . . . 8 |
7 | crest 16081 | . . . . . . . 8 ↾t | |
8 | 4, 6, 7 | co 6650 | . . . . . . 7 ↾t |
9 | 8, 1 | wcel 1990 | . . . . . 6 ↾t |
10 | vy | . . . . . . . . . 10 | |
11 | 10 | cv 1482 | . . . . . . . . 9 |
12 | 11 | csn 4177 | . . . . . . . 8 |
13 | cnei 20901 | . . . . . . . . 9 | |
14 | 4, 13 | cfv 5888 | . . . . . . . 8 |
15 | 12, 14 | cfv 5888 | . . . . . . 7 |
16 | vx | . . . . . . . . 9 | |
17 | 16 | cv 1482 | . . . . . . . 8 |
18 | 17 | cpw 4158 | . . . . . . 7 |
19 | 15, 18 | cin 3573 | . . . . . 6 |
20 | 9, 5, 19 | wrex 2913 | . . . . 5 ↾t |
21 | 20, 10, 17 | wral 2912 | . . . 4 ↾t |
22 | 21, 16, 4 | wral 2912 | . . 3 ↾t |
23 | ctop 20698 | . . 3 | |
24 | 22, 3, 23 | crab 2916 | . 2 ↾t |
25 | 2, 24 | wceq 1483 | 1 𝑛Locally ↾t |
Colors of variables: wff setvar class |
This definition is referenced by: isnlly 21272 nllyeq 21274 |
Copyright terms: Public domain | W3C validator |