Path-connected implies connected

This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., path-connected space) must also satisfy the second topological space property (i.e., connected space)
View all topological space property implications | View all topological space property non-implications
Get more facts about path-connected space|Get more facts about connected space

Statement

If a topological space is a path-connected space, it is also a connected space.

Definitions used

Connected space

Further information: connected space

A topological space is termed connected if it cannot be expressed as a disjoint union of two nonempty open subsets.

Path-connected space

Further information: path-connected space

A topological space $X$ is termed path-connected if, for any two points $x,y\in X$, there exists a continuous map $f$ from the unit interval $[0,1]$ to $X$ such that $f(0)=x$ and $f(1)=y$.

Related facts

Converse

The converse is not true, i.e., connected not implies path-connected.

However, it is true that connected and locally path-connected implies path-connected.

Proof

Fill this in later