Suppose that a topological space $X$ has the property that every continuous real-valued function on $X$ takes on a minimum value. I need to show that any topological space that is homeomorphic to $X$ also possesses this property.
We don't know anything else about the topological space other than this.
Intuitively, it makes sense to me because homeomorphisms preserve open sets, and thus topological properties. It seems to me that boundedness should be a topological property.
However, I have no idea where to even begin with proving this. Please help!
Let $Y$ be a topological space, homeomorphic to $X$. Let $\phi:X\to Y$ be an homeomorphism. Take any continuous function $f:Y\to \Bbb R$. Then $f\circ\phi$ is a continuous, real valued function from $X$, and it takes on a minimum value, but since $f\circ\phi$ and $f$ take on the same values, $f$ also takes on a minimum value (the same).