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Math Help - discrete metric space n singleton sets

  1. #1
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    discrete metric space n singleton sets

    "show that in a discrete metric space, singleton sets are the only nonempty connected sets"


    i have no idea how to go about this one =/
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by cp05 View Post
    "show that in a discrete metric space, singleton sets are the only nonempty connected sets"


    i have no idea how to go about this one =/
    A discrete metric space gives rise to the discrete topology. So, let M be the metric space in common. Clearly each singleton is connected. But, given any m,m'\in M clearly \{m\}\cup\{m'\}=\{m,m'\} is a separation of \{m,m'\} since each \{m\},\{m'\}\subseteq M are open.
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  3. #3
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    oh wow that was so obvious lol thanks so much!
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