when r is =, <, >, m, mi, o, oi, s, si, f, fi, d, or di respectively. In practice, however, fewer normalized spanning intervals are needed, often only one.

The intuition behind the above definition is as follows. Let us handle each of the cases separately.

```
r =< For any intervals i
                      0
                      
                      1 and i
                           0
                            
                           2 in the extensions of i1 and i2 respectively we want i
                                                                         0
                                                                          
                                                                         1 < i
                                                                              0
                                                                              
                                                                              2. From (2)

```

we get r1 (:1^:2) q2. Furthermore, from (14) we get k1 1 r1. Combining these we get k1 (:1^:2^1) q2. In this case, both 2 and 2 are free indicating that either endpoint of i 0 2 can be open or closed.

```
r => For any intervals i
                      0
                      
                      1 and i
                           0
                            
                           2 in the extensions of i1 and i2 respectively we want i
                                                                         0
                                                                          
                                                                         1 > i
                                                                              0
                                                                              
                                                                              2. From (3)

```

we get q1 (:1^:2) r2. Furthermore, from (14) we get q1 Æ1 j1. Combining these we get

```
r2 (:1^:2^Æ1) j1. In this case, both 2 and 2 are free indicating that either endpoint of i
                                                                                      0
                                                                                       
                                                                                      2
                                                                                       

```

can be open or closed.

r = m For any intervals i 0 1 and i 0 2 in the extensions of i1 and i2 respectively we want i 0 1 m i 0 2.

From (4) we get r1 = q2 and 1 6= 2. Furthermore, from (14) we get k1 1 r1 1 l1.

Combining these we get k1 1 q2 1 l1 and 1 6= 2. In this case, only 2 is free indicating that the upper endpoint of i 0 2 can be open or closed.

r = mi For any intervals i 0 1 and i 0 2 in the extensions of i1 and i2 respectively we want i 0 1 mi i 0 2.

From (5) we get q1 = r2 and 1 6= 2. Furthermore, from (14) we get i1 1 q1 Æ1 j1.

Combining these we get i1 1 r2 Æ1 j1 and 1 6= 2. In this case, only 2 is free indicating that the lower endpoint of i 0 2 can be open or closed.

```
r = o For any intervals i
                      0
                      
                      1 and i
                           0
                            
                           2 in the extensions of i1 and i2 respectively we want i
                                                                          0
                                                                          
                                                                          1 o i
                                                                              0
                                                                              
                                                                              2. From (6)

```

we get q1 (1^:2 ) q2 (1^2) r1 (:1^2) r2. Furthermore, from (14) we get i1 1 q1

```
and k1 1 r1 1
                
                 l1. Combining these we get i1 (1^:2^
                                                       1) q2 (1^2^1) l1 and

k1 (:1^2^1) r2. In this case, both 2 and 2 are free indicating that either endpoint of i
                                                                                0
                                                                                 
                                                                                2
                                                                                 

```

can be open or closed.

```
r = oi For any intervals i
                       0
                        
                       1 and i
                              0
                              
                              2 in the extensions of i1 and i2 respectively we want i
                                                                                 0
                                                                                 
                                                                                 1 oi i
                                                                                      0
                                                                                       
                                                                                      2.

```

From (7) we get q2 (:1^2 ) q1 (1^2) r2 (1^:2) r1. Furthermore, from (14) we get

```
r1 1
      
       l1 and i1 
                  1 q1 Æ1 j1. Combining these we get i1 (1^2^
                                                                   1) r2 (1^:2^1) l1

```

and q2 (:1^2^Æ1) j1. In this case, both 2 and 2 are free indicating that either endpoint of i 0 2 can be open or closed.

```
r = s For any intervals i
                      0
                      
                      1 and i
                           0
                            
                           2 in the extensions of i1 and i2 respectively we want i
                                                                          0
                                                                          
                                                                          1 s i
                                                                              0
                                                                              
                                                                              2. From (8)

```

we get q1 = q2, 1 = 2, and r1 (:1^2) r2. Furthermore, from (14) we get i1 1 q1 Æ1 j1 and k1 1 r1. Combining these we get 1 = 2, i1 1 q2 Æ1 j1, and

```
k1 (:1^2^1) r2. In this case, only 2 is free indicating that the upper endpoint of i
                                                                             0
                                                                             
                                                                             2 can

```

be open or closed.

```
r = si For any intervals i
                       0
                        
                       1 and i
                              0
                              
                              2 in the extensions of i1 and i2 respectively we want i
                                                                                 0
                                                                                 
                                                                                 1 si i
                                                                                      0
                                                                                       
                                                                                      2.

```

From (9) we get q1 = q2, 1 = 2, and r1 (1^:2) r2. Furthermore, from (14) we get

```
i1 
      1 q1 Æ1 j1 and r1 1
                                
                                 l1. Combining these we get 1 = 2, i1 
                                                                                  1 q2 Æ1 j1, and

r2 (1^:2^1) l1. In this case, only 2 is free indicating that the upper endpoint of i
                                                                                  0
                                                                                   
                                                                                  2 can

```

be open or closed.