$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Self-defined math definitions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Math symbol commands
\newcommand{\intd}{\,{\rm d}} % Symbol 'd' used in integration, such as 'dx'
\newcommand{\diff}{{\rm d}} % Symbol 'd' used in differentiation
\newcommand{\Diff}{{\rm D}} % Symbol 'D' used in differentiation
\newcommand{\pdiff}{\partial} % Partial derivative
\newcommand{DD}[2]{\frac{\diff}{\diff #2}\left( #1 \right)}
\newcommand{Dd}[2]{\frac{\diff #1}{\diff #2}}
\newcommand{PD}[2]{\frac{\pdiff}{\pdiff #2}\left( #1 \right)}
\newcommand{Pd}[2]{\frac{\pdiff #1}{\pdiff #2}}
\newcommand{\rme}{{\rm e}} % Exponential e
\newcommand{\rmi}{{\rm i}} % Imaginary unit i
\newcommand{\rmj}{{\rm j}} % Imaginary unit j
\newcommand{\vect}[1]{\boldsymbol{#1}} % Vector typeset in bold and italic
\newcommand{\phs}[1]{\dot{#1}} % Scalar phasor
\newcommand{\phsvect}[1]{\boldsymbol{\dot{#1}}} % Vector phasor
\newcommand{\normvect}{\vect{n}} % Normal vector: n
\newcommand{\dform}[1]{\overset{\rightharpoonup}{\boldsymbol{#1}}} % Vector for differential form
\newcommand{\cochain}[1]{\overset{\rightharpoonup}{#1}} % Vector for cochain
\newcommand{\bigabs}[1]{\bigg\lvert#1\bigg\rvert} % Absolute value (single big vertical bar)
\newcommand{\Abs}[1]{\big\lvert#1\big\rvert} % Absolute value (single big vertical bar)
\newcommand{\abs}[1]{\lvert#1\rvert} % Absolute value (single vertical bar)
\newcommand{\bignorm}[1]{\bigg\lVert#1\bigg\rVert} % Norm (double big vertical bar)
\newcommand{\Norm}[1]{\big\lVert#1\big\rVert} % Norm (double big vertical bar)
\newcommand{\norm}[1]{\lVert#1\rVert} % Norm (double vertical bar)
\newcommand{\ouset}[3]{\overset{#3}{\underset{#2}{#1}}} % over and under set
% Super/subscript for column index of a matrix, which is used in tensor analysis.
\newcommand{\cscript}[1]{\;\; #1}
% Star symbol used as prefix in front of a paragraph with no indent
\newcommand{\prefstar}{\noindent$\ast$ }
% Big vertical line restricting the function.
% Example: $u(x)\restrict_{\Omega_0}$
\newcommand{\restrict}{\big\vert}
% Math operators which are typeset in Roman font
\DeclareMathOperator{\sgn}{sgn} % Sign function
\DeclareMathOperator{\erf}{erf} % Error function
\DeclareMathOperator{\Bd}{Bd} % Boundary of a set, used in topology
\DeclareMathOperator{\Int}{Int} % Interior of a set, used in topology
\DeclareMathOperator{\rank}{rank} % Rank of a matrix
\DeclareMathOperator{\divergence}{div} % Curl
\DeclareMathOperator{\curl}{curl} % Curl
\DeclareMathOperator{\grad}{grad} % Gradient
\DeclareMathOperator{\tr}{tr} % Trace
\DeclareMathOperator{\span}{span} % Span
$$
摘要：
Example 1 Let \(X\) be the subspace \([0,1]\cup[2,3]\) of \(\mathbb{R}\), and let \(Y\) be the subspace \([0,2]\) of \(\mathbb{R}\). The map \(p: X \rightarrow Y\) defined by\[p(x)=\begin{cases}x & \t... 阅读全文
摘要：
Exercise 22.3 Let \(\pi_1: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\) be projection on the first coordinate. Let \(A\) be the subspace of \ 阅读全文
摘要：
Arithmetic operations taught in elementary schools are continuous in the high level topological point of view. This signifies that there is literally no clear boundary between simple and complex, low ... 阅读全文
摘要：
Lemma 21.2 (The sequence lemma) Let \(X\) be a topological space; let \(A \subset X\). If there is a sequence of points of \(A\) converging to \(x\), then \(x \in \bar{A}\); the converse holds if \(X\... 阅读全文
摘要：
In this post, I will summarise several topologies established on the product spaces of \(\mathbb{R}\), i.e. \(\mathbb{R}^n\), \(\mathbb{R}^{\omega}\) and \(\mathbb{R}^J\), as well as their relationshi... 阅读全文
摘要：
In my previous post, I introduced various definitions of matrix norms in \(\mathbb{R}^{n \times n}\) based on the corresponding vector norms in \(\mathbb{R}^n\). Meanwhile, the equivalence of differen... 阅读全文
摘要：
Proof of Theorem 20.3Theorem 20.3 The topologies on \(\mathbb{R}^n\) induced by the euclidean metric \(d\) and the square metric \(\rho\) are the same as the product topology on \(\mathbb{R}^n\).Proof... 阅读全文
摘要：
Theorem 20.4 The uniform topology on \(\mathbb{R}^J\) is finer than the product topology and coarser than the box topology; these three topologies are all different if \(J\) is infinite.Proof: a) Prov... 阅读全文
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
