Hyperspace

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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 the empty set $\emptyset$ to be in $\mathcal{H}$.

Properties of hyperspaces

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

Common hyperspaces

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

  1. The hyperspace of closed subsets: $\mathrm{CL}(X) = \{ A \subset X \colon A$ is nonempty and closed in $X \}$
  2. The hyperspace of closed connected subsets: $\mathrm{CLC}(X) = \{A \in \mathrm{CL}(X) \colon A$ is connected $\}$
  3. The hyperspace of compact subsets: $2^X = \{A \in \mathrm{CL}(X) \colon A$ is compact $\}$
  4. The hyperspace of continua: $\mathrm{C}(X)=2^X \bigcap \mathrm{CLC}(X)$
  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 $\mathrm{diam}_d(A)=\mathrm{sup}\{d(x,y)\colon x,y\in A\}$. The hyperspaces $C_{d,\epsilon}(X)$ are called small-point hyperspaces.