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## Homework Statement

Let X be the set of continuous real-valued functions on [0,1]. Prove that S={g in X: g(t) > 0 for all t} is an open subset.

## Homework Equations

## The Attempt at a Solution

I was thinking of taking an arbitrary t[itex]_{0}[/itex] in g and another function f that is also continuous on [0,1], then finding a radius r > g(t[itex]_{0}[/itex])-f(t[itex]_{0}[/itex]) and making the open ball or tube or whatever with that radius about the function g. Somehow I feel that this is incorrect though.