How to set the global variable in a function for cmake?

You need to use set instead of list to affect the variable in the parent scope.

So replace your list command with:

set(source_list ${source_list} ${file_path} PARENT_SCOPE)

PARENT_SCOPE is only for parent, it won't work if you have other non-parent script that want to see it as well.

You need cache for the true "global-like" variable. In your case, use:

SET(source_list  "${source_list}" CACHE INTERNAL "source_list")

Another approach is to use global properties. Once you set it:

set_property(GLOBAL PROPERTY source_list_property "${source_list}")

you can read it from everywhere:

get_property(source_list GLOBAL PROPERTY source_list_property)

I used in examples above the different names for property (source_list_property) and for variable (source_list). Maybe it is better to use the same name. But point is to use a property as global variables, and not about naming.

Such global properties aren't in cache.

Building upon Maxim Suslov's answer, the following code worked for a similar problem I faced:

set_property(GLOBAL PROPERTY source_list)
function(add_source)
    get_property(tmp GLOBAL PROPERTY source_list)
    foreach(arg ${ARGV})
        set(tmp "${tmp} ${arg}")
    endforeach()
    set_property(GLOBAL PROPERTY source_list "${tmp}")
endfunction(add_source)

add_source(a_file)
add_source(b_file c_file)

get_property(local_prop GLOBAL PROPERTY source_list)
message("list: ${local_prop}")

Function add_source can be called from inside any sub-directory.