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finite and countable discrete spaces
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(Theorem)
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Theorem 1 Suppose $X\neq \emptyset$ is equipped with the discrete topology.
- If $X$ if finite, then $X$ is homeomorphic to $\{1,\ldots, n\}$ for some $n\ge 1$ .
- If $X$ if countable, then $X$ is homeomorphic to
 .
Here, $\{1,\ldots, n\}$ and
 are endowed with the discrete topology (or, equivalently, the subspace topology from
 ).
Proof. The first claim will be proven. If $$ X=\{a_1,\ldots, a_n\} $$ let $\Phi\colon \{1,\ldots, n\} \to X$ be $$ \Phi(i)=a_i,\quad i=1,\ldots, n. $$ Since $\Phi$ is a bijection, it is a homeomorphism.
The proof of the second claim is similar to that of the first. 
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"finite and countable discrete spaces" is owned by matte. [ full author list (4) ]
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Cross-references: proof, homeomorphism, bijection, subspace topology, countable, homeomorphic, finite, discrete topology
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This is version 6 of finite and countable discrete spaces, born on 2005-05-19, modified 2006-09-26.
Object id is 7077, canonical name is FiniteAndCountableDiscreteSpaces.
Accessed 1352 times total.
Classification:
| AMS MSC: | 54-00 (General topology :: General reference works ) |
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Pending Errata and Addenda
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