Compact and Locally Compact Spaces

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Posted on 16 Aug 2022, this text provides information on Syllabus Queries related to Course Queries. Please note that while accuracy is prioritized, the data presented might not be entirely correct or up-to-date. This information is offered for general knowledge and informational purposes only, and should not be considered as a substitute for professional advice.

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manpreet Tuteehub forum best answer Best Answer 2 years ago

I would like to consult with anyone who is reading this post on how do you explain the distinction between compact spaces and locally compact spaces to students who had just completed topology course and is now in functional analysis class. In topology, the syllabus cover only Compact spaces and in Functional analysis class, the students were being approach to locally compact spaces.

I can't explain very well and I just say that for locally compact spaces, it is like a subset of a set where the set is a compact spaces. In other words it is a more specific as compared to compact spaces and I don't feel my response is quite clear. Hope to seek some help on the explanation. Thank You.

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manpreet 2 years ago

You also need to be careful, as there are two possible and used definitions of locally compact: (1) every point has a compact neighbourhood; (2) every point has a base of compact neighbourhoods. The advantages of the second definition are: (i) it is in the spirit of definitions of local properties, and (ii) it is very often this property which is used in practice. The two definitions agree for Hausdorff spaces.

Of course the previous comments are on the ball.


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