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# English as a Foreign Language

Viewed 1410 times 2015-4-27 11:53 |Personal category:Logic Test Preparation|System category:Economy| English

Every metric space is a Hausdorff space, the proof of which is normally left as an exercise for the student.

Why bother? We want to converse senbsibly about metric spaces since optimized metrics are part of the solution to the global resource distribution problem, since that problem is solved in part by optimized metrics in fleet management applied to resources and labour pools. Since we know a massive part of the problem is solved by optimizing metrics, we can start having the conversations of high priority in a straightforward way.

To prove every metric space is a Hausdorff space, we want to know what a Hausdorff space is. Note, in the normal conversational direction, definitions preceed conclusions. However, some of my readers prefer backwards logic in which conclusions are announced first, some time for proof may be allowed, and definitions may be included in the proof per sensibility. I mention this since a recent TOEFL listening excersize let us analyze [in Seory English http://siruijiaoyu.com.cn/] the Economics of the undefined idea of supply side Economics, which was talked about verbosely without saying what it is, how to put the pipes together, and what the solution is. Sadly, many students in TOEFL tests struggling with religious truisms mistakely believe they are struggling with English. For these struggline students, here is normal English:

1. Say the definitions first.
2. Say the conclusions, next,
3. Give proof.

A Hausdorff space is one kind of topological space. In a classical sense, we presume the axiom of infinity is valid since we have not switched to finite state machines on three variable crisp, reliable logic. Besides, infinitely many ways in space are interesting. So we won't be defining the idea of what words are, to define the idea of what space is, however, we will let readers know the idea of what a topology is can be found online, for example, at http://en.wikipedia.org/wiki/Topological_space

In a Hausdorff space, distinct points have distinct neighbourhoods having empty intersections. This is the definition.

Why do Hausdorff spaces matter? All the Hausdorff space filter and net limits uniquely obtain. Furthermore, each distinct point can be equivalently defined as its intersection of closed neighbourhoods. This is very important when optimizing fleet management since it implies, yes, for each job the algorithms will close upon a qualified worker. Similarly, responsible resource consumption obtains its unique optimal solution in a metricised benefit pay per contribution relation applied globally to morph us swiftly back into the real carry capacity.

The English I usually teach in normal, real life is how to form a sensible argument, also known as proof; for example, how to argue every metric space is a Hausdorff space.

Suppose a space T has a metric, IE distance makes sense. [Intermediate readers will have found the definition on the internet already.] We have to presume T is infinite to work in classical English, which is okay since people like having infinite stuff and infinite freedom. Let g <> h be two distinct points in T and let the given metric be written as d(g, h) which is a positive number due to distance being sensible. Define a neighbourhood around each of g and h as the open ball of radius 0.3 times d(g, h) and note these two open balls have empty intersection which suffices to meet the definition of T being a Hausdorff space.

1. What is a ball? It is the round, bouncy thing you probably played with when you were in preschool.
2. What is an open ball? It is a ball without skin.
3. What is a radius? It is half the diameter of a uniform, round shape such as a circle, and in higher dimensions, a ball.
4. What is a neighbourhood? In general, these are the interrelated nearby points cohabitant in balls of small radii, therefore, are normally in shared ecosystems = economies.
5. What is times? It is multiplication, I.E. fast addition.

Reference

Real and Abstract Analysis by Edwin Hewitt and Karl Stromberg, the exercise is on page 62.

(Opinions of the writer in this blog don't represent those of China Daily.)

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