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