In this subsection we formalise the idea of a net by introducing a useful concept called a subdivision of a surface. This is a standard kind of net drawn on a surface, and is defined in terms of vertices, edges and faces. It leads to the idea of the Euler characteristic of the surface.

All surfaces obtained from polygons by identifying edges arise from a net (of sorts) consisting of a single polygonal face, together with the edges and vertices that remain aft
Author(s): The Open University

The idea of orientability is another fundamental concept that we need for the study of surfaces. To illustrate the underlying idea, we consider two familiar surfaces â€“ a cylinder and a MÃ¶bius band.

We can distinguish between a cylinder and a MÃ¶bius band by noticing that every cylinder has an â€˜insideâ€™ and an â€˜outsideâ€™, as shown in Author(s): The Open University

As we stated in Section 1, our aim is to classify surfaces up to homeomorphism. So it is worthwhile spending a little time examining what sorts of transformations of surfaces are homeomorphisms. We shall restrict the description to surfaces in space, as these are easier to deal with, though the result at the end of this subsection applies to all surfaces.

Recall that a homeomorphism between two topological spaces (such as surfaces in space) is a bijection with the property that b
Author(s): The Open University

The final surface that can be obtained by identifying edges of a rectangle is even more complicated. Again it cannot be constructed in three dimensions, so is not a surface in space and is hard to visualise. This time we identify both pairs of opposite edges of the rectangle in opposite directions, as indicated by the arrowheads in Author(s): The Open University

In a surface with boundary, the boundary may consist of more than one piece. For example, the boundary of a cylinder, or of a torus with two holes, consists of two pieces, as shown in Figure 23.

Author(s): The Open University

Examples of surfaces with boundary are a cylinder and a MÃ¶bius band. Other examples are the following:

Surfaces with holes

We can obtain a surface with boundary by taking any surface without boundary and punching some holes in it by removing open discs. For example, Figure 19 shows a sphere with 3
Author(s): The Open University

Benjamin Franklin, Part V
Animated biography of Benjamin Franklin's adult life aimed at elementary school children. This video is part of the Discovery Education series.
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Studying mammals: The insect hunters
From pygmy shrews to armadillos, a wide range of mammals survive on a diet made up largely of insects. Many of these have fascinating adaptations suited to catching or rooting out their prey. In this free course, Studying mammals: The insect hunters, you will learn about these adaptations, along with survival strategies for when food is scarce. This is the second course in the Studying mammals series.Author(s): Creator not set

Monica Masperi

Pratiques innovantes en enseignement - apprentissage, l'exemple d'Innovalangues

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This course will allow you to extend your visual modelling skills through the use of the sign graph diagramming technique. You will explore the dynamic relationship between social, economic and ecological factors whose interdependencies will determine the complex dynamics of the future as it unfolds. Your final task will be to identify points of intervention within this complex system in order to "nudge" the situation towards a more favourable outcome.

This OpenLearn course provides a s
Author(s): The Open University

Interview with the Vice Chancellor, Dr John Hood
Dr John Hood talks about the impact of the global recession on universities and how they themselves will contribute to the solution. He also presents his vision for the future of higher education.
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Design
This free course looks at the process of design, from assessing the complexity of design as an activity to exposing the difficulty in making general conclusions about how designers work. You will be able to identify innovation in a wide variety of designed objects and evaluate the impact of this innovation. First published on Thu
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Products, services and branding
In this free course, Products, services and branding, you will find out about how organisations manage their products and services. This will include learning about how new products are developed and how the existing portfolio of offerings is managed. The particular characteristics of service products will be explained so that you can understand the impact that these features have on how services are managed.Author(s): Creator not set

What do we mean by "family"?
The idea of â€˜familyâ€™ is very powerful in contemporary UK culture and policy. Family lives have been the subject of many anxieties both at the personal and policy levels. How do public debates relate to peopleâ€™s everyday experiences of families? In this unit, you can explore the many attempts at defining â€˜familyâ€™ and why these complex and contradictory meanings are important to us. We begin to unpick questions of power and inequality, to test our everyday assumptions about families, and
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More working with charts, graphs and tables
Your course might not include any maths or technical content but, at some point during your studies, itâ€™s likely that youâ€™ll come across information represented in charts, graphs and tables. Youâ€™ll be expected to know how to interpret this information, and possibly encouraged to present your own findings in this way. This unit will help you to develop the skills you need to do this, and gain the confidence to use them. This unit can be used in conjunction with, and builds on the â€˜Working
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"DMC Upside:" Project Del Mar Fall 2012
Fifteen Del Mar College students in the Drama program's make-up class put into practice the skills and techniques they've learned during the fall 2012 semester and presented their best work as part of Project Del Mar's "The Phenomenal Land of Fantastical Fantasies." This segment of the "DMC Upside" television program was featured in the December 2012 edition, which airs on cable channel 19 in Corpus Christi. The "DMC Upside" is produced by DMC TV.
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Topic 5: Income Taxation and Labor Supply part 1 | Economics 2450A: Public Economics
Raj Chetty Fall 2012
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Systems Thinking
Linda Vanasupa
This set of learning materials is aimed at explaining and fostering the discipline of systems thinking.
Some Rights Reserved

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3XSQ: Apple moves in a new direction
Dec. 13 - The Google Maps app will be compatible with any iPhone or iPod Touch that runs iOS 5.1 or higher, the company said in a blog post.
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Happy Holidays to Accepted Freshmen
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