What is Topology about? A nice accessible beginning viewpoint

Reference: Intuitive Concepts in Elementary Topology by B H Arnold, Dover Publications, Inc. NY.ย 

What is Topology?

It is surprising that a fairly satisfactory description of topology can be obtained by changing “geometry” to “topology” , “geometric” to “topological” .

Let us back track a bit —- Euclidean geometry is the study of certain properties of figures in a plane or in space. Not all properties of a figure are of interest — only the geometric properties. For example, the colour of a triangle is not its geometric properties. But length of sides of a triangles, the measures of its angles, its area are certainly geometric properties. So, which properties are geometric ? Answer: Two figures are said to be congruent if and only if one of them can be placed upon the other so that the two figures exactly coincide.

In the above definition of congruence: the emphasis is on the phrase “can be placed upon.” Let us examine the phrase more closely. How do we “place” a figure? How can we move it? What are we allowed to do to it on the way? In geometry, the movements we are allowed are the rigid motions, (translations, rotations, and reflections) in which the distance between any two points of the figure is not changed. Thus, the geometric properties are those that are invariant under rigid motions.

In topology, the movements we are allowed might be called the elastic motions. We imagine that our figures are made of perfectly elastic rubber and in moving a figure, we can stretch, twist, pull and bend it at pleasure. We are even allowed to cut such a rubber figure and tie in a knot, provided that we later sew up the cut exactly as it was before; that is, so that points which were close together before we cut the figure are close together after the cut is sewed up. However, we must be careful that distinct points in a figure remain distinct; we are not allowed to force the two different points to coalesce into just one point. Two figures are said to be topologically equivalent if one figure can be made to coincide with the other by an elastic motion. The topological properties of a figure are those which are also enjoyed by all topologically equivalent figures. Thus, to a topologist, a coffee cup and a dough nut are the same ! ๐Ÿ™‚

Certainly, any topological property of a figure is also a geometric property of that figure, but many geometric properties are not topological properties. The topological properties of a figure can be only the most basic and fundamental of its geometric properties. In topology, when we do elastic transformations, the properties of the figures based on “metric” are lost. That is, lengths, areas, measures of angles are not preserved. So what is preserved? Let us leave this question for some time and play with our elementary view of topology as rubber sheet geometry.

In fact, it might first appear at first glance that no property is a topological one — that, any property of a figure could be changed by an elastic motion ! ๐Ÿ™‚ Fortunately, this is not the case. For instance, a circle C divides the points of a plane into 3 sets — the points inside the circle, the points on the circle and the points outside the circle. Try to visualize this…squeeze…smash the circle…whatever is in the interior will still be in the interior, whatever is on the circle will still be on the circle and whatever is outside it will remain outside it ! ๐Ÿ™‚ So, this is a topological property of a circle ! ๐Ÿ™‚


Nalin Pithwa


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