5.2.1 Proof

Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.

5.2 The identification topology

Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.

5.1 Identifying edges of a polygon

Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.

4.6.1 Remarks

Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.

4.6 The Classification Theorem

Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.

4.5.2 n-fold toruses

Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.

4.5.1 Surfaces with holes

Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.

4.5 Some general results

Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.

4.4 Historical note on the Euler characteristic

Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.

4.3 The Euler characteristic

Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.

4.2 Subdivisions

Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.

4.1 Nets on surfaces

Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.

3.3 The projective plane

Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.

3.2.1 Remarks

Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.

3.2 Orientability

Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.

3.1.1 Inserting half-twists

Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.

3.1 Surfaces with twists

Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.

7 Part 1: 6 Self-assessment questions

This unit is for designers, engineers, technologists and anyone interested in designing and inventing. It is also for managers and consumers interested in innovation and technical change. The unit will show you how design and innovation can create a more sustainable future. It will also help you understand how innovation comes about and will encourage thinking about environmental and social challenges for the future.

5.11 Sustaining innovation and disruptive innovation

This unit is for designers, engineers, technologists and anyone interested in designing and inventing. It is also for managers and consumers interested in innovation and technical change. The unit will show you how design and innovation can create a more sustainable future. It will also help you understand how innovation comes about and will encourage thinking about environmental and social challenges for the future.

5.10 Radical innovation and incremental innovation

This unit is for designers, engineers, technologists and anyone interested in designing and inventing. It is also for managers and consumers interested in innovation and technical change. The unit will show you how design and innovation can create a more sustainable future. It will also help you understand how innovation comes about and will encourage thinking about environmental and social challenges for the future.