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Definition df-nlly 19069
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 19067 . 2  class 𝑛Locally  A
3 vj . . . . . . . . 9  setvar  j
43cv 1368 . . . . . . . 8  class  j
5 vu . . . . . . . . 9  setvar  u
65cv 1368 . . . . . . . 8  class  u
7 crest 14357 . . . . . . . 8  classt
84, 6, 7co 6089 . . . . . . 7  class  ( jt  u )
98, 1wcel 1756 . . . . . 6  wff  ( jt  u )  e.  A
10 vy . . . . . . . . . 10  setvar  y
1110cv 1368 . . . . . . . . 9  class  y
1211csn 3875 . . . . . . . 8  class  { y }
13 cnei 18699 . . . . . . . . 9  class  nei
144, 13cfv 5416 . . . . . . . 8  class  ( nei `  j )
1512, 14cfv 5416 . . . . . . 7  class  ( ( nei `  j ) `
 { y } )
16 vx . . . . . . . . 9  setvar  x
1716cv 1368 . . . . . . . 8  class  x
1817cpw 3858 . . . . . . 7  class  ~P x
1915, 18cin 3325 . . . . . 6  class  ( ( ( nei `  j
) `  { y } )  i^i  ~P x )
209, 5, 19wrex 2714 . . . . 5  wff  E. u  e.  ( ( ( nei `  j ) `  {
y } )  i^i 
~P x ) ( jt  u )  e.  A
2120, 10, 17wral 2713 . . . 4  wff  A. y  e.  x  E. u  e.  ( ( ( nei `  j ) `  {
y } )  i^i 
~P x ) ( jt  u )  e.  A
2221, 16, 4wral 2713 . . 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 18496 . . 3  class  Top
2422, 3, 23crab 2717 . 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 1369 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  19071  nllyeq  19073
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