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Definition df-nlly 20053
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 𝑛Locally  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  |- 𝑛Locally  A  =  { j  e.  Top  | 
A. x  e.  j 
A. y  e.  x  E. u  e.  (
( ( nei `  j
) `  { y } )  i^i  ~P x ) ( jt  u )  e.  A }
Distinct variable group:    u, j, x, y, A

Detailed syntax breakdown of Definition df-nlly
StepHypRef Expression
1 cA . . 3  class  A
21cnlly 20051 . 2  class 𝑛Locally  A
3 vj . . . . . . . . 9  setvar  j
43cv 1398 . . . . . . . 8  class  j
5 vu . . . . . . . . 9  setvar  u
65cv 1398 . . . . . . . 8  class  u
7 crest 14828 . . . . . . . 8  classt
84, 6, 7co 6196 . . . . . . 7  class  ( jt  u )
98, 1wcel 1826 . . . . . 6  wff  ( jt  u )  e.  A
10 vy . . . . . . . . . 10  setvar  y
1110cv 1398 . . . . . . . . 9  class  y
1211csn 3944 . . . . . . . 8  class  { y }
13 cnei 19684 . . . . . . . . 9  class  nei
144, 13cfv 5496 . . . . . . . 8  class  ( nei `  j )
1512, 14cfv 5496 . . . . . . 7  class  ( ( nei `  j ) `
 { y } )
16 vx . . . . . . . . 9  setvar  x
1716cv 1398 . . . . . . . 8  class  x
1817cpw 3927 . . . . . . 7  class  ~P x
1915, 18cin 3388 . . . . . 6  class  ( ( ( nei `  j
) `  { y } )  i^i  ~P x )
209, 5, 19wrex 2733 . . . . 5  wff  E. u  e.  ( ( ( nei `  j ) `  {
y } )  i^i 
~P x ) ( jt  u )  e.  A
2120, 10, 17wral 2732 . . . 4  wff  A. y  e.  x  E. u  e.  ( ( ( nei `  j ) `  {
y } )  i^i 
~P x ) ( jt  u )  e.  A
2221, 16, 4wral 2732 . . 3  wff  A. x  e.  j  A. y  e.  x  E. u  e.  ( ( ( nei `  j ) `  {
y } )  i^i 
~P x ) ( jt  u )  e.  A
23 ctop 19479 . . 3  class  Top
2422, 3, 23crab 2736 . 2  class  { j  e.  Top  |  A. x  e.  j  A. y  e.  x  E. u  e.  ( (
( nei `  j
) `  { y } )  i^i  ~P x ) ( jt  u )  e.  A }
252, 24wceq 1399 1  wff 𝑛Locally  A  =  { j  e.  Top  | 
A. x  e.  j 
A. y  e.  x  E. u  e.  (
( ( nei `  j
) `  { y } )  i^i  ~P x ) ( jt  u )  e.  A }
Colors of variables: wff setvar class
This definition is referenced by:  isnlly  20055  nllyeq  20057
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