Path connectedness implies connectedness

Let $𝑋$ be a topological space. If $𝑋$ is path-connected, then it is also connected.

check

Let $𝛼 : [0, 1] \to 𝑋$ be a path from $𝑥, 𝑦$ . Since $[0, 1]$ is connected (see Connected subspaces of the real line are intervals), then by the Main theorem of connectedness $𝛼 [0, 1]$ is connected. Therefore if $𝑥$ is path connected to $𝑦$ then $𝑥$ is connected to $𝑦$ . Therefore if $𝑋$ is path-connected, i.e. it has only one path-connected component, then $𝑋$ has only one connected component, i.e. it is connected.

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