A property of a topological space X meaning every pair of points x,y in X can be joined by a continuous map γ:[0,1]→X with γ(0)=x and γ(1)=y; often called path-connected or arcwise connected.

Principle

Path-connectedness requires existence of continuous paths between any two points; it is a stronger condition than connectedness and, in locally path-connected spaces, path components are open and coincide with components.

Demonstration

Example: R^n is path-connected because for any x,y the straight-line map γ(t) = (1−t)x + t y is a continuous path from x to y. The circle S^1 is path-connected via arclength parametrizations. A classical counterexample: the topologist's comb space is connected but not path-connected.

Misapplication

Assuming connectedness implies path-connectedness in all spaces; failing to check continuity of an explicitly given 'path' (e.g., piecewise formulas with jumps) or ignoring domain parametrization that fails at endpoints.

Consequence

Path-connected spaces are connected, and many algebraic-topology invariants (fundamental group, covering space theory) are defined using path-homotopy classes of loops; path-connectedness permits lifting and homotopy arguments that require explicit paths.

Reversal

Path-disconnected but connected examples: spaces that are connected yet the existence of continuous paths between some points fails; extreme reversal is totally path-disconnected spaces where no nontrivial path exists between distinct points.

Boundary

Defined for topological spaces; requires continuous maps from the unit interval. It does not quantify the number of distinct homotopy classes of paths (which relates to fundamental group) and excludes weaker forms like local path-connectedness unless specified.

Semantic Tension

Tension with mere connectedness: path-connectedness provides explicit constructive paths and feeds into homotopy theory, while connectedness is purely separation-based; also relates to local path-connectedness which controls openness of path components.

Synthesis

Path-Connectedness is the requirement that any two points can be joined by a continuous path, a constructive strengthening of connectedness that underpins path-based homotopy and covering space arguments.