Elementary Topology and Applications

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Topology, general - Encyclopedia of Mathematics

Another example along these lines occurs in section 3. While these examples of linguistic imprecision are probably easily fixed in lecture, there is a second issue that I think is more problematic. The important concepts of compactness and connectedness are not introduced until quite late in the text, following the chapters on separation axioms and convergence; it seems clear to me, though, that it is more important for a beginning student in topology to know what a connected or compact space is, then it is for him or her to know the difference between a T 3 and T 4 space.

In fact, I think that the concepts of connectedness and compactness are so fundamental to a good understanding of elementary topology that they should have been introduced initially in the metric space chapter, thus giving the reader adequate time in which to absorb and assimilate these ideas.

Likewise, the important concept of a complete metric space is not discussed until very near the end of the book; this is another topic that could and in my opinion should have been brought up earlier, when metric spaces were first discussed. On the plus side, there are certainly aspects of this book that I liked. Working through them would be a valuable activity for students. Solutions are not provided in the text, and I see no indication of a solutions manual being available.

And of course the very inexpensive price of this book about seven dollars on amazon, as I write this is a huge inducement to use this as a text instead of a book that costs twenty times that price. Whether these positive aspects of the book outweigh some of the concerns expressed previously is, of course, a matter of personal taste; in any event, I think this text definitely merits serious consideration as a possible text for an undergraduate-level text in topology.

And, at this price, it is certainly a book that a student or instructor might just want to buy as a reference.

Elementary Applied Topology, Spring 12222

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Mathematics Stack Exchange works best with JavaScript enabled. In , Felix Hausdorff , generalizing the notion of metric space, coined the term "topological space" and gave the definition for what is now called Hausdorff space. Finally, a further slight generalization in , by Kazimierz Kuratowski , gives the present-day concept of topological space.

However, Listing had already used the word for ten years in correspondence.

Graphical Abstract

The separate status of the topologist , a specialist in topology, was used in in the magazine Spectator. A toroid in three dimensions; A coffee cup and a donut are both topologically indistinguishable from this toroid. Topological spaces show up naturally in mathematical analysis , abstract algebra and geometry. This has made topology one of the great unifying ideas of mathematics. General topology , or point-set topology , defines and studies some useful properties of spaces and maps, such as connectedness , compactness and continuity.

Algebraic topology is a powerful tool to study topological spaces, and the maps between them.


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It associates "discrete", more computable invariants to maps and spaces, often in a functorial way. Ideas from algebraic topology have had strong influence on algebra and algebraic geometry. The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the "way they are connected together".

Elementary Topology and Applications

This result did not depend on the lengths of the bridges, nor on their distance from one another, but only on connectivity properties: which bridges are connected to which islands or riverbanks. Similarly, the hairy ball theorem of algebraic topology says that "one cannot comb the hair on a ball smooth". This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere.

In order to deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on. From this need arises the notion of topological equivalence.

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Formally, two spaces are topologically equivalent if there is a homeomorphism between them. In that case the spaces are said to be homeomorphic , and they are considered to be essentially the same for the purposes of topology.

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