4 added 1 characters in body

Suppose $X=\cup K_n$ is a topological space, $K_n$ is a metrizable subspace in $X$ for every $n \in \omega$, then $X$ is a metrizable space?

In metrizable spacespaces, compactness is equivalent to $\sigma$-compactness?

One more: Is pseudocompactness hereditary with respect to $\sigma$-compact subspace?

3 added 92 characters in body

Suppose $X=\cup K_n$ is a topological space, $K_n$ is a metrizable subspace in $X$ for every $n \in \omega$, then $X$ is a metrizable space?

In metrizable space, compactness is equivalent to $\sigma$-compactness?

One more: Is pseudocompactness hereditary with respect to $\sigma$-compact subspace?

2 edited body; added 2 characters in body

Suppose $X=\cup K_n$ is a topological space, $K_n$ is a metrizable subspace in $X$ for every $n \in \omega$, then $X$ is a metrizable space?

In metrizable space, $compactness$ compactness is equivalent to $\sigam$-compact?\sigma\$-compactness?

1