# Applications of Baire's Threom [duplicate]

In a lecture on Baire's Theorem (for complete metric spaces), I gave, for a rather advanced undergraduate class in Real Analysis (covering the theory of metric spaces and elements of general topology), I realised that all the applications I provided were either straight-forward or could be proved using more elementary tools. For example:

a. $\mathbb Q$ is not complete.

b. $\mathbb R$ is uncountable. (Otherwise, if $\mathbb R=\{x_n\}_{n\in\mathbb N}$, then $\bigcap_{n\in\mathbb N} (\mathbb R\setminus\{x_n\})$ would be dense and empty.)

c. A complete metric, with the property that every point is an accumulation point, is uncountable.

Could you suggest more interesting applications? They do not have to be too easy to prove.

## marked as duplicate by Nate Eldredge, Andrés E. Caicedo, Asaf Karagila♦ 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(); } ); }); }); Oct 23 '14 at 15:32

It can be used to (easily) show that the rational numbers $\mathbb{Q}$ are not a $\mathrm{G}_\delta$-subset of $\mathbb{R}$. This in turn can be used to show that there does not exist a function $f: \mathbb{R} \to \mathbb{R}$ which has exactly the set of rational numbers as its points of continuity!