Hit-and-miss topology

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Let $(X,\tau)$ be a topological space. Consider the hyperspace $\mathrm{CL}(X)$. Let $E \subset X$ and define the "hit sets" $$E^- = \{A \in \mathrm{CL}(X) \colon A \cap E \neq \empty \},$$ which are closed sets having nonempty intersection with $E$ and the "miss sets" $$E^+ = \{A \in \mathrm{CL}(X) \colon A \subset E \},$$ which have empty intersection with $E^c$, so they "miss" the complement of $E$.

Examples of hit-and-miss topologies

  1. The Vietoris topology has a subbase

$$\{U^- \colon U \in \tau\} \bigcup \{V^+ \colon V \in \tau\}.$$

  1. The Fell topology has a subbase

$$\{U^- \colon U \in \tau\} \bigcup \{V^+ \colon X \setminus V \mathrm{\hspace{2pt} is \hspace{2pt} compact}\}.$$

References

Beer, Gerald; Tamaki, Robert K. On hit-and-miss hyperspace topologies. Comment. Math. Univ. Carolin.34 (1993), no. 4, 717--728.