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François G. Dorais
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The space $X\times\beta\mathbb{N}$ is normal if and only if $X$ is normal and $\mathfrak{c}$-paracompact. This follows from results of Morita (Paracompactness and product spaces, MR132525), where he generalizes Dowker's characterization of countable paracompactness.

First note that $X\times\beta\mathbb{N}$ is normal if and only if $X\times K$ is normal for every separable compact Hausdorff space $K$. This is because every separable compact Hausdorff space is a perfect image of $\beta\mathbb{N}$.

Morita's Theorem 2.2 shows that if $X$ is normal and $\mathfrak{c}$-paracompact, then $X \times K$ is normal for every compact Hausdorff space $K$ of weight at most $\mathfrak{c}$. Hence, $X\times K$ is normal for every separable compact Hausdorff space $K$ since these all have weight at most $\mathfrak{c}$.

Morita's Theorem 2.4 shows that a space $X$ is normal and $\mathfrak{c}$-paracompact if (and only if) $X\times[0,1]^{\mathfrak{c}}$ is normal. Since the space $[0,1]^{\mathfrak{c}}$ is a separable compact Hausdorff space, this closes the implication loop.

The space $X\times\beta\mathbb{N}$ is normal if and only if $X$ is normal and $\mathfrak{c}$-paracompact. This follows from results of Morita (Paracompactness and product spaces, MR132525), where he generalizes Dowker's characterization of countable paracompactness.

First note that $X\times\beta\mathbb{N}$ is normal if and only if $X\times K$ is normal for every separable compact Hausdorff space. This is because every separable compact Hausdorff space is a perfect image of $\beta\mathbb{N}$.

Morita's Theorem 2.2 shows that if $X$ is normal and $\mathfrak{c}$-paracompact, then $X \times K$ is normal for every compact Hausdorff space of weight at most $\mathfrak{c}$. Hence, $X\times K$ is normal for every separable compact Hausdorff space since these all have weight at most $\mathfrak{c}$.

Morita's Theorem 2.4 shows that a space $X$ is normal and $\mathfrak{c}$-paracompact if (and only if) $X\times[0,1]^{\mathfrak{c}}$ is normal. Since the space $[0,1]^{\mathfrak{c}}$ is a separable compact Hausdorff space, this closes the implication loop.

The space $X\times\beta\mathbb{N}$ is normal if and only if $X$ is normal and $\mathfrak{c}$-paracompact. This follows from results of Morita (Paracompactness and product spaces, MR132525), where he generalizes Dowker's characterization of countable paracompactness.

First note that $X\times\beta\mathbb{N}$ is normal if and only if $X\times K$ is normal for every separable compact Hausdorff space $K$. This is because every separable compact Hausdorff space is a perfect image of $\beta\mathbb{N}$.

Morita's Theorem 2.2 shows that if $X$ is normal and $\mathfrak{c}$-paracompact, then $X \times K$ is normal for every compact Hausdorff space $K$ of weight at most $\mathfrak{c}$. Hence, $X\times K$ is normal for every separable compact Hausdorff space $K$ since these all have weight at most $\mathfrak{c}$.

Morita's Theorem 2.4 shows that a space $X$ is normal and $\mathfrak{c}$-paracompact if (and only if) $X\times[0,1]^{\mathfrak{c}}$ is normal. Since the space $[0,1]^{\mathfrak{c}}$ is a separable compact Hausdorff space, this closes the implication loop.

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François G. Dorais
  • 44.4k
  • 6
  • 150
  • 233

The space $X\times\beta\mathbb{N}$ is normal if and only if $X$ is normal and $\mathfrak{c}$-paracompact. This follows from results of Morita (Paracompactness inand product spaces, MR132525), whichwhere he generalizes Dowker's characterization of countable paracompactness.

First note that $X\times\beta\mathbb{N}$ is normal if and only if $X\times K$ is normal for every separable compact Hausdorff space. This is because every separable compact Hausdorff space is a properperfect image of $\beta\mathbb{N}$.

Morita's Theorem 2.2 shows that if $X$ is normal and $\mathfrak{c}$-paracompact, then $X \times K$ is normal for every compact Hausdorff space of weight at most $\mathfrak{c}$. Hence, $X\times K$ is normal for every separable compact Hausdorff space since these all have weight at most $\mathfrak{c}$.

Morita's Theorem 2.4 shows that a space $X$ is normal and $\mathfrak{c}$-paracompact if and(and only if) $X\times[0,1]^{\mathfrak{c}}$ is normal. Since the space $[0,1]^{\mathfrak{c}}$ is a separable compact Hausdorff space, this closes the implication loop.

The space $X\times\beta\mathbb{N}$ is normal if and only if $X$ is normal and $\mathfrak{c}$-paracompact. This follows from results of Morita (Paracompactness in product spaces, MR132525), which generalizes Dowker's characterization of countable paracompactness.

First note that $X\times\beta\mathbb{N}$ is normal if and only if $X\times K$ is normal for every separable compact Hausdorff space. This is because every separable compact Hausdorff space is a proper image of $\beta\mathbb{N}$.

Morita's Theorem 2.2 shows that if $X$ is normal and $\mathfrak{c}$-paracompact, then $X \times K$ is normal for every compact Hausdorff space of weight at most $\mathfrak{c}$. Hence, $X\times K$ is normal for every separable compact Hausdorff space since these all have weight at most $\mathfrak{c}$.

Morita's Theorem 2.4 shows that a space $X$ is normal and $\mathfrak{c}$-paracompact if and only if $X\times[0,1]^{\mathfrak{c}}$ is normal. Since the space $[0,1]^{\mathfrak{c}}$ is a separable compact Hausdorff space, this closes the implication loop.

The space $X\times\beta\mathbb{N}$ is normal if and only if $X$ is normal and $\mathfrak{c}$-paracompact. This follows from results of Morita (Paracompactness and product spaces, MR132525), where he generalizes Dowker's characterization of countable paracompactness.

First note that $X\times\beta\mathbb{N}$ is normal if and only if $X\times K$ is normal for every separable compact Hausdorff space. This is because every separable compact Hausdorff space is a perfect image of $\beta\mathbb{N}$.

Morita's Theorem 2.2 shows that if $X$ is normal and $\mathfrak{c}$-paracompact, then $X \times K$ is normal for every compact Hausdorff space of weight at most $\mathfrak{c}$. Hence, $X\times K$ is normal for every separable compact Hausdorff space since these all have weight at most $\mathfrak{c}$.

Morita's Theorem 2.4 shows that a space $X$ is normal and $\mathfrak{c}$-paracompact if (and only if) $X\times[0,1]^{\mathfrak{c}}$ is normal. Since the space $[0,1]^{\mathfrak{c}}$ is a separable compact Hausdorff space, this closes the implication loop.

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François G. Dorais
  • 44.4k
  • 6
  • 150
  • 233

The space $X\times\beta\mathbb{N}$ is normal if and only if $X$ is normal and $\mathfrak{c}$-paracompact. This follows from results of Morita (Paracompactness in product spaces, MR132525), which generalizes Dowker's characterization of countable paracompactness.

First note that $X\times\beta\mathbb{N}$ is normal if and only if $X\times K$ is normal for every separable compact Hausdorff space. This is because every separable compact Hausdorff space is a proper image of $\beta\mathbb{N}$.

Morita's Theorem 2.2 shows that if $X$ is normal and $\mathfrak{c}$-paracompact, then $X \times K$ is normal for every compact Hausdorff space of weight at most $\mathfrak{c}$. Hence, $X\times K$ is normal for every separable compact Hausdorff space since these all have weight at most $\mathfrak{c}$.

Morita's Theorem 2.4 shows that a space $X$ is normal and $\mathfrak{c}$-paracompact if and only if $X\times[0,1]^{\mathfrak{c}}$ is normal. Since the space $[0,1]^{\mathfrak{c}}$ is a separable compact Hausdorff space, this closes the implication loop.