Once closed, the database is in mount state but cannot be opened from there on. You need to shut it down to open it, atleast on 817.
Here's the detail -

SVRMGR> startup ;
ORACLE instance started.
Total System Global Area 156633116 bytes
Fixed Size 75804 bytes
Variable Size 44003328 bytes
Database Buffers 112476160 bytes
Redo Buffers 77824 bytes
Database mounted.
Database opened.
SVRMGR> alter database close;
Statement processed.
SVRMGR> select status from v$instance;
STATUS
-------
MOUNTED
1 row selected.
SVRMGR> alter database open;
alter database open
*
ORA-01531: a database already open by the instance
SVRMGR> select status from v$instance;
STATUS
-------
MOUNTED
1 row selected.
SVRMGR> startup open;
ORA-01081: cannot start already-running ORACLE - shut it down first
SVRMGR> shutdown;
ORA-01109: database not open
Database dismounted.
ORACLE instance shut down.
SVRMGR> startup;
ORACLE instance started.


Here's the fun part..although it says database closed, when u say "alter database open" oracle says db is already open.