Space Siii Not Sii 89Edeb

1. **State the problem:** We need to find a topological space $X$ that satisfies the separation axiom $SIII$ (also known as $T_3$ or regular Hausdorff) but does not satisfy $SII$ (also known as $T_2$ or Hausdorff). 2. **Recall definitions:** - A space is $SII$ (Hausdorff) if for any two distinct points, there exist disjoint open neighborhoods separating them. - A space is $SIII$ (regular Hausdorff) if it is $T_1$ and for any point and closed set not containing it, there exist disjoint open neighborhoods separating them. 3. **Important note:** Usually, $SIII$ implies $SII$ because regular Hausdorff spaces are Hausdorff. So a space satisfying $SIII$ but not $SII$ is unusual or impossible under standard definitions. 4. **Check if such a space exists:** By standard topology, $SIII$ spaces are $SII$. Therefore, no space can satisfy $SIII$ but fail $SII$. 5. **Conclusion:** No such space $X$ exists under standard separation axioms. **Final answer:** There is no topological space $X$ satisfying $SIII$ but not $SII$.