Path-Connected Space is Connected

Theorem
Let $$T$$ be a topological space which is path-connected.

Then $$T$$ is connected.

Proof
Let $$D$$ be the discrete space $$\left\{{0, 1}\right\}$$.

Let $$T$$ be path-connected.

Let $$f: T \to D$$ be a continuous surjection.

Let $$x, y \in T: f \left({x}\right) = 0, f \left({y}\right) = 1$$.

Let $$I \subset \R$$ be the closed real interval $$\left[{0 \,. \, . \, 1}\right]$$.

Let $$g: I \to T$$ be a path from $$x$$ to $$y$$.

Then by Continuity of Composite Mapping it follows that $$f \circ g: I \to D$$ is a continuous surjection, contradicting the connectedness of $$I$$ by definition.

Hence the result.

Note
The converse is not necessarily the case.

It is possible for a topological space to be connected but not path-connected.

For example, Graph of Sine of Reciprocal is connected but not path-connected.