A little problem how "open and close and locked door"

As suggested by the Error information given by your compiler:

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Error (10500): VHDL syntax error at uppgift4a_Moore_2.vhd(42) near text "open"; expecting "(", or an identifier ("open" is a reserved keyword), or unary operator

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"open" is a reserved word in VHDL syntax.

For example, you can change your state_type definition to:

type state_type is (opening_1,closing_2, opened, close)

You can use any name for this state except "open".

Any tutorial on VHDL will give a list of reserved words in VHDL syntax.

When writing code, you should avoid using any of these reserved words as a name of user-defined object, such as signal, variable, entity, architecture, enumerated data types, etc.