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Let $(X,\tau)$ be a topological space. A hyperspace of $(X,\tau)$ is the topological space $(\mathcal{H},\tau_v)$ where $\mathcal{H} \subset \mathcal{P}(X)$ is a collection of subsets of $X$ and $\tau_v$ is the Vietoris topology. Typically we only take closed sets to be in $\mathcal{H}$ and do not allow $\emptyset$ to be in $\mathcal{H}$.

Properties of hyperspaces

  • If $(X,\tau)$ is homeomorphic to $(Y,\sigma)$, then $CL(X)$ is homemorphic to $CL(Y)$.

Common hyperspaces

We interpret the topologies on these spaces to be the induced subspace topology from $\tau_v$.

  1. The hyperspace of closed sets: $CL(X) = \{ A \subset X \colon A$ is nonempty and closed in $X \}$
  2. The hyperspace of closed connected sets: $CLC(X) = \{A \in CL(X) \colon A$ is connected $\}$
  3. The hyperspace of compact sets: $2^X = \{A \in CL(X) \colon A$ is compact $\}$
  4. The hyperspace of continua: $C(X)=2^X \bigcap CLC(X)$
  5. If $X$ is a $T_1$ space then the $n$-fold symmetric product of $X$ is the hyperspace $F_n(X)=\{A \subset X \colon 1 \leq |A| \leq n\}$ where $|A|$ denotes the cardinality of $A$. The space $F_1$ is called the space of singletons.
  6. The hyperspace of finite subsets of $X$ is $F(X) = \displaystyle\bigcup_{n=1}^{\infty} F_n(X)$
  7. Let $X$ be a compactum with metric $d$. For any $\epsilon \geq 0$, let $C_{d,\epsilon}(X)=\{A\in C(X)\colon \mathrm{diam}_d(A)\leq \epsilon \}$, where diam $_d(A)=\sup \{d(x,y)\colon x,y\in A\}$. The hyperspaces $C_{d,\epsilon}(X)$ are called small-point hyperspaces.