Answer to mike's other problems

I think I have solved the problems - by reducing the number of event handlers to an absolute minimum and making the user selection event fire once only. I have assumed that the page will be refreshed after the selection has been processed - thus re-enabling the single-shot .one() event handler. If you don't refresh the page after a selection then you simply need to move the handler code from .one() up into a "click" clause of the .on() handler above it. I have also shortened the initial declaration of $tip_provincia. I don't know what this line did, because I do not think there is a selector called ":none". :- $lists.not(this).find('<ol:none>').hide(); ....so, I have removed it completely. Whilst I was writing some code yesterday evening I stumbled across an article saying that .hover() with two functions has been deprecated in jQuery v1.8, so I used this opportunity to change the .hover() into two .on() clauses instead. ----------------------- Answer to https://forum.jquery.com/topic/please-help-to-correct-program-tool-tip#14737000003572162

HTML

<article>
    <div class="provincia"><!-- start provincia -->
        <span class="tip">Choose city</span>
        <ol>
            <li>                       
                <div class="tipo_provinc">
                    <div class="seleccion_provincias"></div>
                    <ul class="menu_lis"></ul>
                </div>
            </li> 
        </ol>
        <output id="test_select"></output>
    </div><!-- end provincia -->
</article>

CSS

span.provincia_arrow,
a.provincia_tip{
    display:none;
    position:absolute;    
}
a.provincia_tip{     
    height: 23px;
    text-align:center;
    padding:4px 10px 0 10px;
    width: auto;
    background-color:#000;
    border-radius: 5px;
    border: #F00 solid 2px ;
    margin-left:-95px;
    margin-top:-50px;
    color:#FFF;    
    text-decoration:none;     
}
span.provincia_arrow{
    margin-left:-75px;
    margin-top:-25px;     
}
/* START DIV "SELECT PROVINCE" */
div.provincia {/*div superior*/
    font-family: sans-serif;
    font-size: 15px;
    margin-top: 115px;
    margin-left:155px;
}
div.provincia span.tip {    
    display: inline-block;
    min-width: 128px;
    height: 13px;
    color: #000;
    background-color:  #FFF;
    padding: 3px 0 7px 4px;
    cursor: pointer;
    border:3px solid #CCC;    
}
div.provincia ol {/*div inf of "province"*/
    position: absolute;
    display: none;
    border: 1px solid #333;
    width: 96px;
    overflow: hidden;     
    background-color: #000;
    z-index: 10000;
    margin-left:;
    margin-top: -2px;    
    height:100px;    
    border:1px solid #0CF;  
}
div.provincia ol li a {
    margin-left:15px;
    font-size:16px;
    color: #999;    
}
div.provincia ol li a:hover {
    background-color: #666;
    color: #FFF;        
}
div.provincia ol li:last-child {
    border-bottom: none;
}
div.provincia output#test_select {
    position:relative;
    top:150px;
    left:-115px;
}
/* END DIV "SELECT PROVINCE" */

.seleccion_provincias {
    display:none;      
    
    /*this line is commented just to show functionality*/
}

JavaScript

$(document).ready(function () {

    /* cache frequent elements to prevent many DOM searches*/
    var $lists = $('div.provincia ol');
    var $spanTip = $('span.tip');
    var $tip_provincia = $('<a class="provincia_tip">City</a><span class="provincia_arrow">▼</span>');

    /* construct menu */
    var provs = ['Aires', 'Catamarca', 'Chaco', 'Chubut', 'Córdoba'];
    $.each(provs, function (i, v) {
        $('.seleccion_provincias')
            .append('<input type= "radio" name="provincias" value="' + v + '" />');
        $('.menu_lis')
            .append('<li><a href="#" rel="' + i + '">' + v + '</a></li>');
    });

    $('div.provincia').append($tip_provincia);
    
    $spanTip.on({
        mouseenter: function () {
            $tip_provincia.fadeIn('fast');
        },
        mouseout: function () {
            if (!$(this).data('activesong')) {
                $tip_provincia.fadeOut('fast');
            }
        }
    });
    $spanTip.on('click', function () { //on click
        $lists.slideDown('fast');
        $spanTip.data('activesong', true);
    });
    $('.menu_lis li a').on('click', function (e) {
        e.preventDefault();
        e.stopPropagation();
        var $tgt = $(this);
        $spanTip.html($tgt.html());
        $spanTip.data('activesong', false);
        $('#test_select').val('Seleccion: ' + $tgt.attr('rel'));
        $('.seleccion_provincias :radio:eq(' + $tgt.attr('rel') + ')').prop('checked', true);
        $lists.slideUp('fast');
        $tip_provincia.fadeOut('fast');
    });

});