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Definition df-nlly 17209
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 17207 . 2  class 𝑛Locally  A
3 vj . . . . . . . . 9  set  j
43cv 1631 . . . . . . . 8  class  j
5 vu . . . . . . . . 9  set  u
65cv 1631 . . . . . . . 8  class  u
7 crest 13341 . . . . . . . 8  classt
84, 6, 7co 5874 . . . . . . 7  class  ( jt  u )
98, 1wcel 1696 . . . . . 6  wff  ( jt  u )  e.  A
10 vy . . . . . . . . . 10  set  y
1110cv 1631 . . . . . . . . 9  class  y
1211csn 3653 . . . . . . . 8  class  { y }
13 cnei 16850 . . . . . . . . 9  class  nei
144, 13cfv 5271 . . . . . . . 8  class  ( nei `  j )
1512, 14cfv 5271 . . . . . . 7  class  ( ( nei `  j ) `
 { y } )
16 vx . . . . . . . . 9  set  x
1716cv 1631 . . . . . . . 8  class  x
1817cpw 3638 . . . . . . 7  class  ~P x
1915, 18cin 3164 . . . . . 6  class  ( ( ( nei `  j
) `  { y } )  i^i  ~P x )
209, 5, 19wrex 2557 . . . . 5  wff  E. u  e.  ( ( ( nei `  j ) `  {
y } )  i^i 
~P x ) ( jt  u )  e.  A
2120, 10, 17wral 2556 . . . 4  wff  A. y  e.  x  E. u  e.  ( ( ( nei `  j ) `  {
y } )  i^i 
~P x ) ( jt  u )  e.  A
2221, 16, 4wral 2556 . . 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 16647 . . 3  class  Top
2422, 3, 23crab 2560 . 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 1632 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  17211  nllyeq  17213
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