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Definition df-nlly 17522
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 teminology - 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 17520 . 2  class 𝑛Locally  A
3 vj . . . . . . . . 9  set  j
43cv 1651 . . . . . . . 8  class  j
5 vu . . . . . . . . 9  set  u
65cv 1651 . . . . . . . 8  class  u
7 crest 13640 . . . . . . . 8  classt
84, 6, 7co 6073 . . . . . . 7  class  ( jt  u )
98, 1wcel 1725 . . . . . 6  wff  ( jt  u )  e.  A
10 vy . . . . . . . . . 10  set  y
1110cv 1651 . . . . . . . . 9  class  y
1211csn 3806 . . . . . . . 8  class  { y }
13 cnei 17153 . . . . . . . . 9  class  nei
144, 13cfv 5446 . . . . . . . 8  class  ( nei `  j )
1512, 14cfv 5446 . . . . . . 7  class  ( ( nei `  j ) `
 { y } )
16 vx . . . . . . . . 9  set  x
1716cv 1651 . . . . . . . 8  class  x
1817cpw 3791 . . . . . . 7  class  ~P x
1915, 18cin 3311 . . . . . 6  class  ( ( ( nei `  j
) `  { y } )  i^i  ~P x )
209, 5, 19wrex 2698 . . . . 5  wff  E. u  e.  ( ( ( nei `  j ) `  {
y } )  i^i 
~P x ) ( jt  u )  e.  A
2120, 10, 17wral 2697 . . . 4  wff  A. y  e.  x  E. u  e.  ( ( ( nei `  j ) `  {
y } )  i^i 
~P x ) ( jt  u )  e.  A
2221, 16, 4wral 2697 . . 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 16950 . . 3  class  Top
2422, 3, 23crab 2701 . 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 1652 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 set class
This definition is referenced by:  isnlly  17524  nllyeq  17526
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