custom jQuery checkbox allow only one selection

So, the first thing i did was swap out all instances of: checkbox with 'radio' still didn't get my effect.. I want to keep the 'check mark' and custom background of the box etc that this jQuery script allows, everything is perfect, I just need to assure that only ONE ITEM can be checked / selected..

The number one, the jQuery..

/*! custom-checkbox - v1.0.1  */
(function ($) {

    $.fn.customCheckbox = function() {

        return this.each( function(i,v) {

            //Ensure that a checkbox element was passed
            if ( !$(v).is(':checkbox') ) {
                return false;         
            }

            //Add classes
            $(v).addClass("custom-checkbox");

            //If not wrapped within label tags, wrap it
            var parentLabel = $(v).parent("label");
            var withinLabel = parentLabel.length;

            if ( !withinLabel ){
                $(v).wrap("<label class='custom-checkbox-label'></label>");      
            }  
            else {
                parentLabel.addClass('custom-checkbox-label');
            }

            //Create dummy checkbox
            var dummy = $("<span class='custom-checkbox-display'></span>");
            $(v).after(dummy);
            if ( $(v).prop("checked") ) {
                $(v).next('.custom-checkbox-display').addClass("checked");
            }

            //Add/remove classes to checkbox whenever state changes
            $(v).change( function(e) {
                var checkbox = $(e.currentTarget);
                var state = checkbox.prop("checked");
                if ( state ) {
                    dummy.addClass("checked");
                }
                else {
                    dummy.removeClass("checked");
                }
            });

            //Make reset button aware of the custom checkboxes
            var form = $(v).parents("form");
            var reset = form.find("input[type='reset']");
            reset.each( function(ri,rv) {
                if ( !$(rv).hasClass("custom-checkbox-aware") ) {
                    $(rv).addClass("custom-checkbox-aware");
                    $(rv).click( function() {
                        form.find(".custom-checkbox:checked").trigger("click");
                    });
                }
            });

        });

    };

}(jQuery));

html form mark-up...

  <div class="form-group_district">
    <p class="label">District Name*</p><br>
            <label class="labeldistrict">
            <input type="checkbox" name="NYCDistrictName" class="checkboxed" required >
            I teach in the New York City Department of Education</label><br>
            <label class="labeldistrict">
            <input type="checkbox" name="ChiDistrictName" class="checkboxed" required >
            I teach in the Chicago Public Schools</label><br>
            <label class="labeldistrict"> 
            <input type="checkbox" name="LADistrictName" class="checkboxed" required >
            I teach in the Los Angeles Unified School District</label><br>
            <label class="labeldistrict">
            <input type="checkbox" name="othDistrictName" class="checkboxed" required >
            Other</label><br>               
            <input  type="text" name="othDistrictName" maxlength="100" class="otherdistrict" required ><br>
            <br>
    </div>

For the fiddlers out there.. here is a bare bones demo. just imagine the 'check markup' images were showing on selection / focus. (I don't really want to save them and host them somewhere for this demo).

http://ift.tt/1D1N6ld