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Let $(X,\tau)$ be a [[topological_space | topological space]]. A <em>hyperspace</em> 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 | Vietoris topology]]. Typically we only take closed sets to be in $\mathcal{H}$ and do not allow $\emptyset$ to be in $\mathcal{H}$.
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Welcome to the Hyperspace Wiki! This wiki is for people who study [[Hyperspace | Hyperspace]]s, [[Continuum Theory]], and, more generally, [[Topology]]. Please [http://hyperspacewiki.org/index.php/Special:CreateAccount create an account] and [[hyperspacewiki:Todo|contribute to the Wiki]]. See our section on [[Problems]] such as a full transcription of [[The Houston Problem Book]], and follow our [https://twitter.com/hyperspacewiki Twitter].
  
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=Properties of hyperspaces=
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{| class="wikitable"
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* If $(X,\tau)$ is [[homeomorphism | homeomorphic]] to $(Y,\sigma)$, then $CL(X)$ is homemorphic to $CL(Y)$.
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|-
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! scope="row" colspan="2" | Registration
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| Due to a resurgence of automated spam bots, account registration and anonymous editing is currently <b style="color:red;">disabled</b>. Please contact Tom Cuchta ([mailto:tomcuchta@gmail.com tomcuchta@gmail.com]) to gain access to edit the wiki.
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|}
  
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=Common hyperspaces=
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== Overview ==
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We interpret the topologies on these spaces to be the induced subspace topology from $\tau_v$.
 
  
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# The <em>hyperspace of closed sets</em>: $CL(X) = \{ A \subset X \colon A$ is nonempty and closed in $X \}$
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{| class="wikitable" style="text-align: center; width:60%; margin:auto;"
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# The <em>hyperspace of closed connected sets</em>: $CLC(X) = \{A \in CL(X) \colon A$ is connected $\}$
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! colspan="6" | Hyperspace Wiki
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# The <em>hyperspace of [[compact]] sets</em>: $2^X = \{A \in CL(X) \colon A$ is compact $\}$
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|-
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# The <em>hyperspace of continua</em>: $C(X)=2^X \bigcap CLC(X)$
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| [[Hyperspace Theory]]
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# If $X$ is a $T_1$ space then the <em>$n$-fold symmetric product of $X$</em> 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 <em>space of singletons</em>.
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| Overview of Hyperspace Theory
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# The <em>hyperspace of finite subsets of $X$</em> is $F(X) = \displaystyle\bigcup_{n=1}^{\infty} F_n(X)$
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|-
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# Let $X$ be a compactum with metric $d$. For any $\epsilon \geq 0$, let $C_{d,\epsilon}(X)=\{A\in C(X)\colon $ 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 <em>small-point hyperspaces</em>.
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| [[Continuum Theory]]
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| Overview of Continuum Theory
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|-
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| [[Topology]]
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| Overview of Topology
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|-
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| [[The Houston Problem Book]]
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| Problems and Solutions for the Houston Problem Book
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|-
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| [[Set Theory]]
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| Overview of Set Theory and Set-Theoretic Topology
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|-
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| [[hyperspacewiki:Community|Community Portal]]
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| Community Portal
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|-
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| [[hyperspacewiki:Citation Needed| Master Citations Needed List]]
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| Master List of Pages With Bibliographies That Require Attention
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|-
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| [[hyperspacewiki:Todo | Master Task List]]
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| Master List of Pages With Tasks That Require Editor Attention
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|-
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| [[Help:Main|Help Portal]]
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| Help Portal
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|-
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| [[Special:Categories|Categories]]
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| Overview of the Current Topic Categories in the Wiki
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|}
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== Newest Pages ==
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{{Special:NewPages}}
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== Recent Changes ==
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{{Special:Recentchanges}}
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== External links ==
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# [http://wikiwiki.jp/compactness/?%A6%D3-pseudocompact Encyclopedia of Compactness Wiki*]<br />
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# [http://www.karlin.mff.cuni.cz/~pyrih/e/ Examples in Continuum Theory]<br />
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# [http://web.mst.edu/~wjcharat/JJC/publications/p138.pdf History of Continuum Theory]<br />
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# [http://web.mst.edu/~continua/ Open problems in continuum theory]<br />
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# [http://web.mst.edu/~ingram/hpb.pdf University of Houston Mathematics Problem Book (10/14/94)]<br />
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# [http://www.karlin.mff.cuni.cz/~pyrih/open/htm/problem.1.html Pavel Pyrih Problem Book]<br />
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== References ==
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<references />

Latest revision as of 02:50, 24 December 2018

Welcome to the Hyperspace Wiki! This wiki is for people who study Hyperspaces, Continuum Theory, and, more generally, Topology. Please create an account and contribute to the Wiki. See our section on Problems such as a full transcription of The Houston Problem Book, and follow our Twitter.

Registration Due to a resurgence of automated spam bots, account registration and anonymous editing is currently disabled. Please contact Tom Cuchta (tomcuchta@gmail.com) to gain access to edit the wiki.

Overview

Hyperspace Wiki
Hyperspace Theory Overview of Hyperspace Theory
Continuum Theory Overview of Continuum Theory
Topology Overview of Topology
The Houston Problem Book Problems and Solutions for the Houston Problem Book
Set Theory Overview of Set Theory and Set-Theoretic Topology
Community Portal Community Portal
Master Citations Needed List Master List of Pages With Bibliographies That Require Attention
Master Task List Master List of Pages With Tasks That Require Editor Attention
Help Portal Help Portal
Categories Overview of the Current Topic Categories in the Wiki

Newest Pages

Recent Changes

No changes during the given period match these criteria.

External links

  1. Encyclopedia of Compactness Wiki*
  2. Examples in Continuum Theory
  3. History of Continuum Theory
  4. Open problems in continuum theory
  5. University of Houston Mathematics Problem Book (10/14/94)
  6. Pavel Pyrih Problem Book

References

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