REFERENCE SHEET

Topological spaces & homotopy

Study properties preserved by continuous deformation rather than by a particular distance.

Before this Sets & maps, Limits & continuity

H(x,0)=f(x), H(x,1)=g(x)

Definitions

Topology

A family of open subsets containing ∅ and X, closed under arbitrary unions and finite intersections.

Continuous map and homeomorphism

A map is continuous when preimages of open sets are open. A homeomorphism is a continuous bijection whose inverse is continuous.

Homotopy

A continuous H:X×[0,1]→Y interpolating between f and g as displayed. A homotopy equivalence has a continuous inverse up to homotopy, not necessarily an actual inverse.

Key theorems

Compact-to-Hausdorff criterion

A continuous bijection from a compact space to a Hausdorff space is a homeomorphism.

Path connected implies connected

If any two points can be joined by a continuous path, the space cannot be split into two disjoint nonempty open subsets. The converse fails in general.

Deformation retraction

If X deformation retracts onto A, then the inclusion A→X is a homotopy equivalence.

Notable applications

Shape simplification

Replace a complicated space by a homotopy-equivalent simpler one before computing invariants.

Data and geometry

Separate properties of connectivity and holes from the choice of coordinates or scale.

Work it out

Try before revealing

01 / Construct

Give a deformation retraction of ℝ² without the origin onto the unit circle.

Hint

Move each point along its radial line.

Solution

H(x,t)=((1−t)+t/‖x‖)x for x≠0. The coefficient is positive, the endpoint lies on the circle, and circle points stay fixed.

02 / Explain

Why is a nonempty convex subset of ℝⁿ contractible?

Hint

Choose a point a in it and use straight lines.