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What I want to know is this:

Am I correct in thinking that if we have a finite topological space (so no reals or anything here), that any connected subset contained in this space has only one element? For example, {x} is connected but {x,y} is disconnected.

user138798
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1 Answers1

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No; as a simple case, if any space $X$ has the indiscrete topology (only the empty set and $X$ are open), then any subset of $X$ is connected.

In more detail: Suppose $V \subset X$. Then the relative topology on $V$ is also the indiscrete topology. Thus $V$ has no open subsets other than the empty set and $V$, and is therefore connected.

Ted
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  • I am so sorry to have to ask you again but this topic in analysis is flying over my head. If I take a subset of X and it is open, and has two elements, then why can I not form a separation with those two elements? – user138798 Mar 29 '14 at 05:46
  • @user138798 If $X$ has the indiscrete topology, there are no open subsets of $X$ other than $\emptyset$ and $X$. See edits. – Ted Mar 29 '14 at 05:52
  • Okay, thank you so much. Just bear with me for one last thing. I now understand what the subsets of X are. Does it matter what elements X has? – user138798 Mar 29 '14 at 06:05
  • @user138798 No it doesn't (other than trivialities, like $X$ having one element or no elements) - the indiscrete topology is defined the same way for every set. –  Mar 29 '14 at 06:09