Confusion about Connected sets and Connected components

Question

I was reading the definition of Connected Sets and Connected Components from Wikipedia and was having trouble understanding the definition.

Consider the specific example set: $S = \{y:y \leq |x|\} \in R^2$. I managed to prove that it is a connected set. But what are the connected components of this set? I am not sure if it is the set itself or we can split it up into $S_1 = \{y: y < -x, x < 0\}$ and $S_2 = \{y: y \leq x, x \geq 0\}$ are these connected components?

Can someone explain to me what exactly connected components are? The definition is very confusing.

Answer 1

A subsets $S$ of a topological space $X$ is a connected component of $X$ if and only if it is connected and it is a maximal element of the family of connected susets of $X$, ordered by inclusion $\subseteq$. Theis means that $S$ is connected, and that $S'$ is a connected subset such that $S'\supseteq S$ if and only if $S'=S$. If $X$ is a connected topological space, then the one and only connected component of $X$ is $X$ itself. In your example, neither of those sets $S_i$ is a connected component because neither of them sets is maximal by inclusion among the connected subsets of $S$.

Answer 2

No, that's a completely different matter! For example, if $y= 9- x^2$, a continuous function, then $\{y| y< 0\}$ is the nonconnected union of the two separated sets $y< -3$ and $y> 3$.