Remove container from text widget

CMS & WordPress
1

Is there a function to remove the inner div that gets inserted when using a text widget?
In the functions.php to register the widget areas I use

if ( function_exists('register_sidebar') )
register_sidebar(array(
'name' => 'Page Top',
'id' => 'Page Top Box',
'description' => 'Page Top Box',
'before_widget' => '<div class="page-top">',
'after_widget' => '</div>',
'before_title' => '<h2 class="page-top-title">',
'after_title' => '</h2>',
));

Using the text widget this is what renders as code

<div class="page-top"><h2 class="page-top-title">About Me</h2>
<div class="textwidget"><p>Nam posuere felis ac urna. Vestibulum tempor vestibulum urna. Nullam metus. Vivamus ac purus. Nullam interdum ullamcorper libero. Morbi vehicula imperdiet justo. Etiam mollis fringilla ante. Donec et dui. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos hymenaeos. Etiam mi libero, luctus nec, blandit ac, rutrum ac, lectus.</p></div>
</div>

I want to remove the div (<div class=“textwidget”>) from around the widget. In my searching, all I have found is a hack to the core file wp-includes>default-widgets.php. I don’t want to hack core files that will probably get updated at some point. The function in the default-widgets.php file that creates the div is

function widget( $args, $instance ) {
	extract($args);
	$title = apply_filters( 'widget_title', empty($instance['title']) ? '' : 	$instance['title'], $instance, $this->id_base);
	$text = apply_filters( 'widget_text', $instance['text'], $instance );
	echo $before_widget;
	if ( !empty( $title ) ) { echo $before_title . $title . $after_title; } ?>
	<div class="textwidget"><?php echo $instance['filter'] ? wpautop($text) : $text; ?></div>
	<?php
	echo $after_widget;
}
2

I guess there’s not an answer here. WordPress.org doesn’t have an answer either except to hack the core file which is not exceptable to me as an answer.

3

Isn’t there a way to pass empty strings as before_widget and after_widget arguments?

4

If I understand this correctly, I’m already passing before and after in the sidebar registration

'before_widget' => '<div class="page-top">',
'after_widget' => '</div>',

The problem is the widget inserts another div inside around the content which is not needed and actually has no value.
I simply want to know if there’s a function that will strip out the div.