# Is there a name for a topological space $X$ in which every proper closed subset is compact? [duplicate]

Is there a name for a topological space $X$ in which every proper closed subset is compact$^{(*)}$? It is well known that in a compact topological space, every closed set is compact. Hence, the class of compact spaces is contained in the class of spaces with $(*)$ property.
## marked as duplicate by user99914, Noah Schweber, Martin Sleziak, Henno Brandsma general-topology StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Jun 14 '16 at 6:03
• The total space $X$ should be closed in any topology, so what you find is just a compact space. – cjackal Jun 14 '16 at 5:13