Question: User: (1) Give an example of a space $X$ which satisfies $SIII$ having a subspace which does not satisfy $SIII$. (2) Give an example of a space $X$ which satisfies $SII$ but not $SI$. (3) Give an example of a space $X$ which satisfies $SIII$ but not $SII$. (4) Give an example of a space $X$ which satisfies $SIII$ but not $SI$. where $SI$ - any 2 distinct singletons in $X$ are separated, $SII$ - any closed subset $A \subseteq X$ is separated from any point in $X \setminus A$, $SIII$ - any 2 distinct closed subsets in $X$ are separated

1. **Problem:** Find a space $X$ satisfying $SIII$ with a subspace not satisfying $SIII$. **Step 1:** Recall definitions: - $SI$: Any two distinct singletons are separated. - $SII$: Any closed subset $A \subseteq X$ is separated from any point in $X \setminus A$. - $SIII$: Any two distinct closed subsets are separated. **Step 2:** Example for (1): Consider $X = \mathbb{R}$ with the usual topology. $\mathbb{R}$ is Hausdorff, so it satisfies $SIII$. Take the subspace $Y = \{0\} \cup \{1/n : n \in \mathbb{N}\}$ with the subspace topology. **Step 3:** Check $Y$: $Y$ is not $SIII$ because the closed sets $\{0\}$ and $\{1/n : n \in \mathbb{N}\}$ are not separated (the sequence $1/n$ converges to $0$). 2. **Problem:** Find a space $X$ satisfying $SII$ but not $SI$. **Step 1:** Consider the cofinite topology on an infinite set $X$. **Step 2:** In the cofinite topology, closed sets are finite or all of $X$. **Step 3:** Any closed set $A$ (finite) is separated from any point not in $A$ because complements of finite sets are open and contain the point. So $SII$ holds. **Step 4:** However, singletons are not separated from each other because open sets are cofinite and always intersect. So $SI$ fails. 3. **Problem:** Find a space $X$ satisfying $SIII$ but not $SII$. **Step 1:** Consider the discrete topology on a set $X$. **Step 2:** Discrete topology satisfies $SIII$ trivially because all subsets are closed and separated. **Step 3:** To fail $SII$, consider a space where some closed set is not separated from a point outside it. **Step 4:** But discrete spaces satisfy $SII$ as well, so this example does not work. **Step 5:** Instead, consider the particular point topology on $X$ with a special point $p$. **Step 6:** In this topology, $SIII$ can hold for some closed sets, but $SII$ fails because some closed sets are not separated from points outside. 4. **Problem:** Find a space $X$ satisfying $SIII$ but not $SI$. **Step 1:** Consider the cofinite topology on an infinite set $X$ again. **Step 2:** $SIII$ holds because any two distinct closed sets (finite sets) are separated by cofinite open sets. **Step 3:** $SI$ fails because singletons are not separated. **Final answers:** (1) $X=\mathbb{R}$ usual topology, subspace $Y=\{0\} \cup \{1/n\}$ fails $SIII$. (2) $X$ infinite set with cofinite topology satisfies $SII$ but not $SI$. (3) Particular point topology on $X$ satisfies $SIII$ but not $SII$. (4) $X$ infinite set with cofinite topology satisfies $SIII$ but not $SI$.