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can anybody please explain me what is meant by "the components" of space?

How can we determine the components of a space?

What are the components of a cofinite space and cocountable space?

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    Path connected components? Connected components?2012-12-21

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Let $X$ be a space. A subset $C$ of $X$ is a connected component of $X$ if (1) $C$ is connected, and (2) whenever $C\subsetneqq Y\subseteq X$, the set $Y$ is not connected. In other words, $C$ is a maximal connected subset of $X$.

Define a relation $\sim$ on $X$ by saying that $x\sim y$ if and only if there is a connected set $C\subseteq X$ that contains both $x$ and $y$. It’s not too hard to prove that $\sim$ is an equivalence relation, and that the $\sim$-equivalence classes are the connected components of $X$. This implies that the connected components of a space form a partition of the space into maximal connected subsets. And since the closure of a connected set is connected, each of these maximal connected subsets (i.e., components) of $X$ must be closed.

If $X$ has the cofinite topology, every infinite subset of $X$ is connected. Assuming that $X\ne\varnothing$, there are two possibilities.

  1. If $X$ is finite, its components are its singletons, i.e., its one-point subsets.
  2. If $X$ is infinite, $X$ is connected, so its only component is $X$ itself.

If $X$ has the co-countable topology, every uncountable subset of $X$ is connected, and there are again two possibilities.

  1. If $X$ is countable, its components are its singletons, i.e., its one-point subsets.
  2. If $X$ is uncountable, $X$ is connected, so its only component is $X$ itself.
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    @ccc: I did, and I’m typing up some of the answers right now.2012-12-22