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Definition df-nlly 19762
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 19760 . 2  class 𝑛Locally  A
3 vj . . . . . . . . 9  setvar  j
43cv 1378 . . . . . . . 8  class  j
5 vu . . . . . . . . 9  setvar  u
65cv 1378 . . . . . . . 8  class  u
7 crest 14676 . . . . . . . 8  classt
84, 6, 7co 6284 . . . . . . 7  class  ( jt  u )
98, 1wcel 1767 . . . . . 6  wff  ( jt  u )  e.  A
10 vy . . . . . . . . . 10  setvar  y
1110cv 1378 . . . . . . . . 9  class  y
1211csn 4027 . . . . . . . 8  class  { y }
13 cnei 19392 . . . . . . . . 9  class  nei
144, 13cfv 5588 . . . . . . . 8  class  ( nei `  j )
1512, 14cfv 5588 . . . . . . 7  class  ( ( nei `  j ) `
 { y } )
16 vx . . . . . . . . 9  setvar  x
1716cv 1378 . . . . . . . 8  class  x
1817cpw 4010 . . . . . . 7  class  ~P x
1915, 18cin 3475 . . . . . 6  class  ( ( ( nei `  j
) `  { y } )  i^i  ~P x )
209, 5, 19wrex 2815 . . . . 5  wff  E. u  e.  ( ( ( nei `  j ) `  {
y } )  i^i 
~P x ) ( jt  u )  e.  A
2120, 10, 17wral 2814 . . . 4  wff  A. y  e.  x  E. u  e.  ( ( ( nei `  j ) `  {
y } )  i^i 
~P x ) ( jt  u )  e.  A
2221, 16, 4wral 2814 . . 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 19189 . . 3  class  Top
2422, 3, 23crab 2818 . 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 1379 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  19764  nllyeq  19766
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