Product of Countable Discrete Space with Sierpiński Space is Paracompact

Theorem

Let $T_X = \struct {S, \tau}$ be a countable discrete space.

Let $T_Y = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space.

Let $T_X \times T_Y$ be the product space of $T_X$ and $T_Y$.

Then $T_X \times T_Y$ is paracompact.

Proof

From Discrete Space is Paracompact, $T_X$ is paracompact.

We have that the Sierpiński space $T_Y$ is a finite topological space.

From Finite Topological Space is Compact, $T_Y$ is a compact space.

This needs considerable tedious hard slog to complete it. In particular: Steen and Seebach in Part $\text I$ chapter $3$ Compactness: Invariance Properties offer "If $X$ is compact, then in general $X \times Y$ has the compactness properties of $Y$." Whether this holds for $Y$ paracompact needs to be investigated. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $11$. Sierpinski Space: $19$