Variables have scope. If you're trying to use a variable inside a function it will be local to the function. To use one from outside the function you need to declare it as global.
function someFunc() {
global $DEFAULT_TOP_PAGE_ID;
// more code...
$this->top_page_ID = $DEFAULT_TOP_PAGE_ID;
// etc.
}
In this case I think you probably need a definition:
define("DEFAULT_TOP_PAGE_ID", "top_");
then
function someFunc() {
// more code...
$this->top_page_ID = DEFAULT_TOP_PAGE_ID;
// Note: $ has gone ^ here
// etc.
}