SQL Server 2014

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CREATE TABLE [Cust](
        [Cust_id] [varchar](10) NULL,
        [Open_Date] [date] NULL,
        [Closed_Date] [date] NULL
)

insert into [Cust] values ('a123','10/01/2019','10/15/2019')
insert into [Cust] values ('a123','10/12/2019','11/01/2019')
insert into [Cust] values ('b245','07/01/2019','09/15/2019')
insert into [Cust] values ('b245','10/12/2019','12/01/2019')
4 rows affected

select 
    cust_id,
    min(open_date) open_date,
    max(closed_date) closed_date
from (
    select 
        t.*,
        sum(case when not open_date <= lag_closed_date then 1 else 0 end) 
            over(partition by cust_id order by open_date) grp
    from (
        select 
            t.*,
            lag(closed_date) over (partition by cust_id order by open_date) lag_closed_date
        from cust t
    ) t
) t
group by cust_id, grp


cust_id
open_date
closed_date


a123
2019-10-01
2019-11-01

b245
2019-07-01
2019-09-15

b245
2019-10-12
2019-12-01


WITH x (cust_id, open_date, closed_date, lvl, grp) AS (
   SELECT cust_id, open_date, closed_date, 1, 1
     FROM (
             SELECT cust_id
                  , open_date
                  , closed_date
                  , row_number()
                    OVER (PARTITION BY cust_id ORDER BY closed_date DESC, open_date) AS rn
               FROM cust
          ) AS t
    WHERE rn = 1
    UNION ALL
   SELECT cust_id, open_date, closed_date, lvl, grp
     FROM (
             SELECT c.cust_id
                  , c.open_date
                  , c.closed_date
                  , x.lvl + 1                                                              AS lvl
                  , x.grp + CASE WHEN c.closed_date < x.open_date THEN 1 ELSE 0 END        AS grp
                  , row_number() OVER (PARTITION BY c.cust_id ORDER BY c.closed_date DESC) AS rn
               FROM cust c
                    JOIN x
                         ON x.cust_id = c.cust_id
                            AND c.open_date < x.open_date
          ) AS t
    WHERE t.rn = 1
     )
SELECT cust_id, min(open_date) AS first_open_date, max(closed_date) AS last_closed_date
  FROM x
 GROUP BY cust_id, grp
 ORDER BY cust_id, grp


cust_id
first_open_date
last_closed_date


a123
2019-10-01
2019-11-01

b245
2019-10-12
2019-12-01

b245
2019-07-01
2019-09-15
cust_id	open_date	closed_date
a123	2019-10-01	2019-11-01
b245	2019-07-01	2019-09-15
b245	2019-10-12	2019-12-01