 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.
Author(s): The Open University

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.
Author(s): The Open University

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.
Author(s): The Open University

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.
Author(s): The Open University

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.
Author(s): The Open University

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.
Author(s): The Open University

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.
Author(s): The Open University

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.
Author(s): The Open University

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.
Author(s): The Open University

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.
Author(s): The Open University

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.
Author(s): The Open University

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.
Author(s): The Open University

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.
Author(s): The Open University

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.
Author(s): The Open University

The first step is to develop a picture (called in soft systems terminology a rich picture) that encapsulates all the elements that people think are involved in the problem. Once the rich picture has been drawn, the analyst will attempt to extract ‘issues’ and key tasks.

Issues are areas of contention within the problem situation. Key tasks are the essential jobs that must be undertaken within the problem situation.

Author(s): No creator set

The approach begins with a situation in which one or more people perceive that there is a problem. It will not be possible to define the problem or its setting with any precision and, in any event, the different people involved will have different ideas.

Various ‘softer’ approaches to problem solving have been proposed. The one that I shall describe is based on (although not exactly the same as) the methodology developed by Peter Checkland and his collaborators at the University of Lancaster. This has been applied to systems problems in a number of projects.

The soft systems approach is based on a number of key principles.

• Problems do not have an existence that is independent of the peo
Author(s): No creator set

Implementation involves all the detailed design, development and installation tasks required to get the agreed proposal operating.

Figure 34 shows an arrow leading from ‘implementation’ to ‘problem/opportunity’; this recognises that implementation is never the end of the story. The successful completion of a project will give rise either to other opportunities or to a further set of problems that ne
Author(s): No creator set

You might imagine that after all that has gone before, the decision about whether to go ahead or not would be automatic, but this is rarely the case. There will still be much discussion and ‘fine tuning’ necessary to ensure that the proposal is acceptable. It is at this stage that any qualitative measures of performance are brought into play.

While the identified objectives and constraints have been referred to constantly during the development stage, the testing stage of the approach is a more formal analysis of each option. Its objective is to determine whether:

• the option will meet the operational objectives

• it is technically feasible

• it is organisationally feasible

• it will meet the financial objectives.

Author(s): No creator set