Always use strict and warnings.  With that, you'd have gotten warnings like Variable "$open" will not stay shared, which might have been a hint that there's still something wrong with the scoping of the variable $open.

The problem is in particular that you've declared $open locally to the callback function push_button, and that you've declared the other callback functions locally to that function (which I had only realized after having indented your code reasonably...).   Perl does not have local subroutines, and if you declare named subs within other subs, they're essentially parsed as if they were declared top-level. Hence the scope issue.

Here's a stripped down version (with "minor" edits to make it actually runnable...), which uses real global variables for $open and $te (for which the same issues apply), so that they can be shared across various callback routines.

#!/usr/bin/perl -w
use strict;
use Tk;

my $open;
my $te;

# Entry Window
my $mw = new MainWindow;

# The START button:
my $button = $mw -> Button(-text => "Start",-command=>\&push_button)->
+pack();

MainLoop;

sub push_button {
    my $mw = new MainWindow;
    $mw -> geometry ("1200x600");

    # Menu bar
    
    $mw->configure(-menu => my $menubar = $mw->Menu,
                           -background => 'white');
    my $file = $menubar->cascade(-label => '~File');
    my $edit = $menubar->cascade(-label => '~Edit');
    my $help = $menubar->cascade(-label => '~Help');
    
    $mw-> Button (-text =>'Open', -command =>\&open_file)->place(-x=>2
+40, -y=>35); 
    $mw-> Button (-text =>'Get Statistics', -command =>\&get_statistic
+s)->place(-x=>320, -y=>35); 
    
    $te = $mw->Scrolled('TextUndo')->place(-x=>240, -y=>70);

    MainLoop;
}

sub open_file {
    $open = $mw->getOpenFile(
                #-filetypes => $types_OPEN,
                -defaultextension => '.sff'
            );
    $te->Load($open);
}

sub get_statistics { 
    my $fastafile= $open;
    print STDERR "selected file: $fastafile\n";
}
[download]

A few more comments:

• Indenting your code in some sane way already gets you a long way to better understanding it.
• Always try to provide code samples that actually run. That way you significantly increase the chances of people looking into your problem. In fact, I only went to the trouble of editing your sample code because I just happened to have a few idle minutes.
• I think (but I'm no Tk expert) that having another MainLoop within a callback function is somewhat "unusual" style.
• And, at the risk of repeating myself: use strictures/warnings — they have been invented to help you.

I do not really know how to thank you, you have been very kind ! Now all works ! My apologies for not posting my question well. It's my first programming experience so i I have a lot to learn. Heartfelt thanks for all your help !! Cheers, Giorgio

Hi Eliya, following all your directions i corrected the script code (with the help of 'warnings' and 'strict'), now i have no much error messages coming out. Further, i was able to integreate inside the script of %statistics without problems (thanks to you surely)! Do you know if it's possible to print the result in a window ? (messagebox, or a little scrolled window, etc..). For now the result is printed out in the terminal and i would like to see it in a GUI window.

#all global variable set well !

........


$mw-> Button (-text =>'Open', -command =>\&open_file)->place(-x=>240, 
+-y=>35); 

$mw-> Button (-text =>'Get Statistics', -command =>\&get_statistics)->
+place(-x=>320, -y=>35); 








sub open_file {$open = $mw->getOpenFile(

                               -filetypes => $types_OPEN,

                              -defaultextension => '.sff');

                                      $te->Load( "$open");}






sub get_statistics { 
my $fastafile= $open;

    open (FASTA, "<$fastafile");
$/ = ">";

my $junkFirstOne = <FASTA>;

while (<FASTA>) {
    chomp;
    my ($def,@seqlines) = split /\n/, $_;
    my $seq = join '', @seqlines;
    $stat->add_data(length($seq));
}

%distrib = $stat->frequency_distribution(\@bins);
print "Total reads:\t" . $stat->count() . "\n";
print "Total nt:\t" . $stat->sum() . "\n";
print "Mean length:\t" . $stat->mean() . "\n";
print "Median length:\t" . $stat->median() . "\n";
print "Mode length:\t" . $stat->mode() . "\n";
print "Max length:\t" . $stat->max() . "\n";
print "Min length:\t" . $stat->min() . "\n";
print "Length\t# Seqs\n";
foreach (sort {$a <=> $b} keys %distrib) {
    print "$_\t$distrib{$_}\n";}};

#This is the result in the terminal for an example $open file:

Total reads:    4981
Total nt:    104724
Mean length:    21.024693836579
Median length:    21
Mode length:    21
Max length:    27
Min length:    18
Length    # Seqs
18    290
19    456
20    874
21    1974
22    626
23    281
24    373
25    69
26    32
27    6
28    0
[download]

It works fine and correctly. Do you know how to print this result in a GUI windows and not only in the terminal? Thanks you !!!

Do you know if it's possible to print the result in a window ?

Sure, see Perl tk - open file and print content