(X)HTML container always consist of an opening an a closing tag. Although earlier standards allowed for many of the closing tags to be omitted, newer (XML based) standards do not, which is reason enough to make it a habit of always closing elements with the appropriate closing tag.
Besides that, even forgiving browsers may sometimes have a hard time to figure out what you had in mind structure-wise; using closing tags tells them exactly, avoiding rendering surprises and potentially slower rendering.
Finally, closing your container elements properly will help yourself lateron to better understand what document structure you had in mind back in the time you made it.
General rule of thumb is to close all tags that you can and give your code as much information as you can. Don't let the browser guess what you mean - TELL IT WHAT YOU WANT!
Close everything!
Also with <p> tags, if you use a stylesheet and add a class to a paragraph it will not stop until it reaches a </p>
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