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Be $X$ a set of cofinite topology and $Y \subseteq X$. Are $\tau_{c}$ and $\tau_{s}$, respectively, the cofinite topology in $Y$ and the topology of $Y$-subspace induced by the topology of $X$. Compare $\tau_{c}$ and $\tau_{s}$ with respect inclusion.

Definition 1. Be $X$ an infinite set and set $\tau=\{\emptyset\}\cup \{V\subseteq X:X-V$ is finite$\}$.

Definition 2. Be $X$ a set, $\tau$ a topology, and $Y\subseteq X$. Then the $X$-induced topology is given by: take the $X$ open and intersect with $Y$.

Can anyone help me with this? I think the only difference from $\tau_{c}$ topology and topology is that in $\tau_{c}$ topology: \begin{equation} \tau_{c}=\{\emptyset\}\cup \{V\subseteq Y: Y-V \text{ is finite } \} \end{equation}

while whereas

\begin{equation} \tau_{s}=\{\emptyset\}\cup \{V\subseteq X: X-V \text{ is finite } \}\cap Y \end{equation}

in my view it doesn't change anything. Change anything, which is the finest?

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    $\begingroup$ You are correct: they are the same topology. $\endgroup$ – Brian M. Scott Dec 22 '20 at 19:33
  • $\begingroup$ Brian, How can I prove that fact? $\endgroup$ – Isadora Suhadolnick Dec 22 '20 at 19:39
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Take a finite set $V\subset Y$, this means $Y- V$ is open in the cofinite topology of $Y$, now $X-V$ is also open in the cofinite topology of $X$ and $Y\cap (X-V)=Y-V$, so $Y- V$ is also open in the topology inherited by $X$. It is pretty obvious that $Y\cap (X- V)$ is open in the cofinite topology of $Y$ for any finite $V\subset X$, so you have the double incusion.

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If $V\in\tau_c$ let $F=Y\setminus V$, and let $U=X\setminus F$. $F$ is a finite subset of $X$, so $U$ is open in $X$, and clearly

$$U\cap Y=(X\setminus F)\cap Y=(X\cap Y)\setminus F=Y\setminus F=V\,,$$

so $V\in\tau_s$. This shows that $\tau_c\subseteq\tau_s$; now see if you can show that $\tau_s\subseteq\tau_c$ to conclude that $\tau_s=\tau_c$.

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  • $\begingroup$ okay, thank you!!!!! $\endgroup$ – Isadora Suhadolnick Dec 22 '20 at 20:01
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    $\begingroup$ @IsadoraSuhadolnick: You’re welcome! $\endgroup$ – Brian M. Scott Dec 22 '20 at 20:08

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