topology products, unions and bijections

1. Show that the y-axis is a closed subset of $\mathbb{R}_{fc} \times \mathbb {R}_{fc}$ ($\mathbb{R}_{fc}$ is the finite complement topology on $\mathbb{R}_{fc}$)

2. Prove false: every continuous bijection has a continuous inverse.

3. Show that if X and Y are Hausdorff, then so is their disjoint union X $\bigcup^{*}$*Y*

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What have you tried? –  Stefan Hamcke Nov 29 '12 at 17:57
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1 Answer

HINTS:

1. $\Bbb R\setminus\{0\}$ and $\Bbb R$ are open sets in $\Bbb R_{fc}$.

2. Look at the simplest possible bijection from $\Bbb R_{fc}$ to $\Bbb R$ with the usual topology.

3. You have to consider three cases: separating two points of $X$, separating two points of $Y$, and separating a point of $X$ from a point of $Y$. For all three cases it’s relevant that $X$ and $Y$ are open subsets of their disjoint union.

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