Stretching Without Breaking

Topology · Axiom Academy

INTRO Stretching Without Breaking Exploring continuous maps: the functions that preserve connections Imagine you have a rubber band. You can stretch it, twist it, compress it—but as long as you don't tear it, the band remains fundamentally the same object. Every point stays connected to its neighbors. In topology, functions that work like this are called continuous maps . They transform spaces while preserving their essential structure: if two points are close before the transformation, they stay close after. Continuous Transformation in Action Watch as a square continuously transforms into a circle. Notice how: Points move smoothly—no sudden jumps Nearby points stay nearby throughout The shape morphs continuously—no tearing or breaking What Makes a Map Discontinuous? A map is discontinuous if it breaks connections—if it tears, jumps, or creates gaps. Let's compare the two side by side: Stretches smoothly—preserves all connections Creates a tear—breaks the structure Experience Continuous Deformation Now it's your turn! Drag the control points below to create your own continuous deformations. No matter how much you stretch and warp the shape, the continuous map keeps everything connected. Drag the blue control points to deform the grid continuously

This is the written version of the interactive lesson above. See the full Topology course.