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This post summarises different ways of constructing continuous functions, which are introduced in Section 18 of James Munkres “Topology”.Constant function.Inclusion function.N.B. The function domain s... 阅读全文
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Theorem 18.4 in James Munkres “Topology” states that if a function \(f : A \rightarrow X \times Y\) is continuous, its coordinate functions \(f_1 : A 阅读全文
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Understanding of continuity definition in topology When we learn calculus in university as freshmen, we are usually force-fed with the \(\epsilon-\del 阅读全文
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Theorem 16.3 If \(A\) is a subspace of \(X\) and \(B\) is a subspace of \(Y\), then the product topology on \(A \times B\) is the same as the topology 阅读全文
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According to Wikipedia, the well known barber paradox states like this:The barber is the “one who shaves all those, and those only, who do not shave themselves.” The question is, does the barber shave... 阅读全文
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The definition of topological basis for a space $X$ requires that each point $x$ in $X$ is contained in one of the said topological bases. Meanwhile, 阅读全文
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In this article, I’ll present my ideas about how to improve mathematics study, which are the forewords of my handwriting notes on Royden’s “Real Analy 阅读全文
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Make mathematics study a habit with dogged perseverance Don't build mansion on top of loose sand. Concrete a solid foundation by allocating at least o 阅读全文
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Exercise 1. Let \(X\) be a space. Let \(\mathcal{D}\) be a collection of subsets of \(X\) that is maximal with respect to the finite intersection prop 阅读全文
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(Remark: The proof presented in this post is a reorganization and interpretation of that given by James Munkres in his book "Topology".) Theorem 37.3 阅读全文