12 2018 档案
摘要：Theorem 19.6 Let \(f: A \rightarrow \prod_{\alpha \in J} X_{\alpha}\) be given by the equation\[f(a) = (f_{\alpha}(a))_{\alpha \in J},\]where \(f_{\alpha}: A \rightarrow X_{\alpha}\) for each \(\alpha...
阅读全文
摘要：In James Munkres “Topology”, the concept for a tuple, which can be \(m\)-tuple, \(\omega\)-tuple or \(J\)-tuple, is defined from a function point of view as below.Let \(X\) be a set.Let \(m\) be a pos...
阅读全文
摘要：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
摘要：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
摘要：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
摘要：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
摘要：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
摘要：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
浙公网安备 33010602011771号