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Definition df-nlly 19838
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 19836 . 2  class 𝑛Locally  A
3 vj . . . . . . . . 9  setvar  j
43cv 1380 . . . . . . . 8  class  j
5 vu . . . . . . . . 9  setvar  u
65cv 1380 . . . . . . . 8  class  u
7 crest 14692 . . . . . . . 8  classt
84, 6, 7co 6278 . . . . . . 7  class  ( jt  u )
98, 1wcel 1802 . . . . . 6  wff  ( jt  u )  e.  A
10 vy . . . . . . . . . 10  setvar  y
1110cv 1380 . . . . . . . . 9  class  y
1211csn 4011 . . . . . . . 8  class  { y }
13 cnei 19468 . . . . . . . . 9  class  nei
144, 13cfv 5575 . . . . . . . 8  class  ( nei `  j )
1512, 14cfv 5575 . . . . . . 7  class  ( ( nei `  j ) `
 { y } )
16 vx . . . . . . . . 9  setvar  x
1716cv 1380 . . . . . . . 8  class  x
1817cpw 3994 . . . . . . 7  class  ~P x
1915, 18cin 3458 . . . . . 6  class  ( ( ( nei `  j
) `  { y } )  i^i  ~P x )
209, 5, 19wrex 2792 . . . . 5  wff  E. u  e.  ( ( ( nei `  j ) `  {
y } )  i^i 
~P x ) ( jt  u )  e.  A
2120, 10, 17wral 2791 . . . 4  wff  A. y  e.  x  E. u  e.  ( ( ( nei `  j ) `  {
y } )  i^i 
~P x ) ( jt  u )  e.  A
2221, 16, 4wral 2791 . . 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 19264 . . 3  class  Top
2422, 3, 23crab 2795 . 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 1381 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  19840  nllyeq  19842
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