Connected not implies path-connected

This article gives the statement and possibly, proof, of a non-implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., Connected space (?)) need not satisfy the second topological space property (i.e., Path-connected space (?))
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Statement

It is possible for a topological space to be a connected space but not a path-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 is termed path-connected if, given any two distinct points in the topological space, there is a path from one point to the other. Here, a path is a continuous function from the unit interval to the space, with the image of $0$ being the starting point or source and the image of $1$ being the ending point or terminus.

Partial truth

The following are true:

• For subsets of the real line (with the usual topology), the notions of connected and path-connected coincide. Specifically, the connected sets, which are also the path-connected sets, are precisely the intervals (open, closed, half-open half-closed, and possibly extending to infinity in one or both directions). Note that from our examples, it is clear that this breaks down for subsets of $\mathbb {R} ^{2}$.
• For locally path-connected spaces, the notions of connectedness and path-connectedness coincide. This includes open subsets of Euclidean space, locally Euclidean spaces, and in particular, manifolds.

Converse

The converse is true, i.e., path-connected implies connected.

Proof

Examples of the topologist's sine curve

Further information: topologist's sine curve, closed topologist's sine curve

The topologist's sine curve is the union of the graph of the function $\sin(1/x)$ for $x$ in the interval $(0,1]$ and the origin. (There are other variants -- for instance, the right endpoint is not always taken as $1$ but can be any positive number). It acquires the subspace topology from the Euclidean plane.

• The topologist's sine curve is not path-connected: There is no path connecting the origin to any other point on the space.
• The topologist's sine curve is connected: All nonzero points are in the same connected component, so the only way it could be disconnected is if the origin and the rest of the space were the two connected components. But in that case, both the origin and the rest of the space would be open subsets, and the origin is not open in the space becaues an arbitrarily small ball around the origin intersects the rest of the space.

The closed topologist's sine curve is the closure, in the Euclidean plane, of the topologist's sine curve. It includes all points on the $y$-axis with $y$-coordinate between $-1$ and $1$. For reasons similar to the above, the closed topologist's sine curve is connected but not path-connected.

Example of the infinite broom

Further information: infinite broom

The infinite broom is another example of a topological space that is connected but not path-connected.

Note that unlike the case of the topologist's sine curve, the closure of the infinite broom in the Euclidean plane, known as the closed infinite broom (also sometimes as the broom space) is a path-connected space.