Talk:Relationship between Component Types

Utterly false claims
Let $X = \{0\}\times [0,1] \cup \{(x,\sin \frac 1 x): x > 0 \}$ have the subspace topology as a subspace of $\R^2$.

Then $X$ is connected, but not path connected, as is well known.

Let $C$ be the path component of $(0,0)$.

Then $C$ is NOT a component of $X$, because $C ≠ X$ and the only component of $X$ is $X$. Every single other statement on the page is wrong for similar reasons. If you want to talk about the relationship between sets of arc components, path components, etc., then you need to replace "is a subset of" with "is a refinement of" throughout. --Dfeuer (talk) 22:58, 7 June 2013 (UTC)