UPDATE:

I've implemented most of the suggestions I received and the little program now looks like this:

#!/usr/bin/perl -w
#Purpose: To Take a DOS file wildcard and thus take all the matching c
+ustom SGML files in the working directory and subdivide them into new
+ files whose names are the id's of the divs in those original files.

use strict;

print "What file(s) do you want to run this program on?\n";
our $lines = "";
our @InFileNames;
our @OutFileNames;
our @OutFileContent;
my $i = 0;
my $j = 0;
my $TheFile = <STDIN>;
chomp ($TheFile);

#open file and get all text
sub OpenFile {
open(FILE, $_[0]) or $lines = "";
local $/ = undef;
$lines = <FILE>;
#remove blank lines
$lines =~ s/\n{2}/\n/gms;
close(FILE);
}


#add ¥ to closing div tags
sub MarkClose {
$lines =~ s/(<\/div>)/$1¥/gms;
}

#open output.txt for appending and write results to it
sub FileAppend {
my $Outfile =  ">>" .  $_[0] . ".bsd";
my $Content = $_[1];
open(FILE, $Outfile) or die "Can't open $Outfile.\n";
print FILE $Content;
print FILE "\n";
close FILE;
}

#Create an array containing all file in the directory matching the glo
+b.
sub GetInFileList {
my $FileDef = $_[0];
@InFileNames = glob($FileDef);
}

#Populate an array with the contents of the id attribute of every <div
+> tag in the input file.
sub GetOutFilesList {
my $OutFile;
@OutFileNames = $lines =~ m/<div[^>]*>/gms;
foreach $OutFile (@OutFileNames){
    $OutFile =~ s/<div type=[^\s]* id="([^>]*)">/$1/gms;
    $OutFile =~ s/\./_/gms;
}
}

#Subdivides the File into the subfiles.
sub GetOutFileContent {
my $LinesString = $_[0];
@OutFileContent = split /¥/,  $LinesString;
}

### Does the job
&GetInFileList($TheFile);
for ($i = 0, $i < @InFileNames, $i++) {
    &OpenFile($InFileNames[$i]);
    &MarkClose();
    &GetOutFilesList;
    &GetOutFileContent($lines);
    for ($j = 0, $j < @OutFileNames, $j++){
        &FileAppend($OutFileNames[$j], $OutFileContent[$j]);
    }
}

#be nice and say it's done
print "Program Finished\n";
[download]

The script now dies at a very specific point: On line 13 or possibly 18, where it gives the following error message:

Use of uninitialized value in open at ##program name censored## line 18, <STDIN> line 1.

I take this to mean that the array entitled @InFileNames has no contents, because the glob function used to fill it on line 45 didn't behave as I thought it would.

Also, the debug mode behaved oddly when it reached the <STDIN> line, it prompted me for input with the line DB(1), and then prompted me again with DB(2) when I had given it its input, and so on ad infinitum.

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