Question Related to $G_\delta$ and $F_\sigma$ I have question related to these two statements.


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*If $X$ and $Y$ are homeomorphic subsets of the real line $R$ and $X$ is an $F_\sigma$-set in $\mathbb{R}$, then $Y$ is an $F_\sigma$-set, too.

*If $X$ is a Tikhonov space, every Cech-complete subspace of $X$ is $G_\delta$-set.


I know the second statement is wrong but I have not counter-example. Also I want to know the outline proof of the first one.
 A: *

*Use that a subset of the real line is $F_\sigma$ if and only if it is $\sigma$-compact. 

*Take $\omega_1+1=[0,\omega_1]$ with the order topology. Then singleton $\{\omega_1\}$ is Cech-complete, but not $G_\delta$.
(Here $\omega_1$ denotes the first uncountable ordinal, and we consider the space of all countable ordinals together with the first uncountable ordinal.) 
Here are some details related to 1. Say,$A=\cup_{n<\omega} F_n$ where each $F_n$ is closed. Then each $F_n$ is the union of countably many compact sets, namely $F_n=\cup_{k\ge1}\Bigl([-k,k]\cap F_n\Bigr)$, where each $\Bigl([-k,k]\cap F_n\Bigr)$ is closed-and-bounded, hence compact. It follows that $A$ is the union of countably many compact sets. Hence, if $f:A\to B$ is a homeomorphism, it follows that $B$ is the union of countably many compact sets. Since each compact subset of the real line is closed, $B$ is the union of countably many closed sets, hence $B$ is $F_\sigma$. 
Details related to 2. Singleton $\{\omega_1\}$ is compact, hence Cech-complete. Take any countable family $\{U_n:n<\omega\}$ of neighborhoods of $\{\omega_1\}$. We may assume that these are basic open neighborhoods, that is there are $\beta_n$ such that $U_n=(\beta_n,\omega_1]$ for each $n$. Let $\beta=\sup_{n<\omega} \beta_n$, then $\beta<\omega_1$ (since sup of any countable set of countable ordinals is countable, since the union of countably many countable sets is countable). But, $\cap\{U_n:n<\omega\}=\cap_{n<\omega}\,(\beta_n,\omega_1] =(\beta,\omega_1] \not= \{\omega_1\}$. (Here $\omega$ is the first infinite ordinal, it is countable, same as $\Bbb N=\{0,1,..\}$ the set of all non-negative integers.) 
