Let $X$ be a topological space Prove that each path component of $X$ is path connected.
This is a homework problem so what I want is a explanation of what this question means not a answer. My question is why is this even a question it seems to follow by definition of path component. Am I missing something?
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$\begingroup$There are various potential ways to define path-component, for example:
Definition: The path-components of $X$ are those path-connected subsets $Y$ such that any $Z$ with $Y \subsetneq Z \subseteq X$ is not path-connected.
Here it is immediate that path-components are path-connected, but unfortunately, it is not immediate that path-components exist in the first place.
Definition: The path-components of $X$ are the sets of the form $\{y \in X \mid \exists$ a path from x to y$\}$, where $x \in X$.
Here the existence is immediate, but proving that path-components are indeed path-connected requires a (very simple) proof.
To make a notion such as path-component really useful, one wants to prove that both definitions actually are equivalent.
$\endgroup$ $\begingroup$So $C=\{y\in X|\text{there exists a path from }x \rightarrow y\}$ also $x \in X$. $C$ is path connected because if we pick $x_1,x_2 \in X$. Then there exits (by def of $C$) a path from $x_1 \rightarrow y$ and a path from $x_2 \rightarrow y$. So by transitivity there exits a path from $x_1 \rightarrow x_2$. So $C$ is path connected.
$\endgroup$ 1 $\begingroup$I think we can just follow the Wiki path componet:
First we define a relation between two points $x$ and $y$ in a topological space $X$.
Then we define that $x \sim y$ if there exists a path between $x$ and $y$, which indues a relation class $[x] = \{y|x\sim y\}$. Then we call $[x]$ a path component of $X$.
And now it directly proves your question.
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