In this lecture, I have give a intrinsic topological view of equivalent norm and equivalent on topological space and some basis about weak topology and weak star topology in topological vector space. however I think there still need some examples about topological equivalent cases to have grasp the essence about this idea.
There are two topologies on X, named $\tau_{1}$,$\tau_{2}. \quad x_{n}\rightarrow$ x in topology $\tau_{1} \Rightarrow x_{n} \rightarrow$ x in topology $\tau_{2}$, but the topology $\tau_{1}$ is not stronger than $\tau_{2}$.
when the two topologies is given by metrics, it is obvious that $\tau_{1}$ is stronger than $\tau_{2}$
Considering the discrete topology and co-finite topology, it is clear that co-finite topology is weaker than topology, for the fact that discrete topology is the strongest topology on X. however the convergence of sequence is the same on both two topologies. for any more details about co-finite topology one can scan
Xu senlin‘s book “set theory“.
we can not distinguish such two different topologies by sequence convergence trick, for further improvement of sequence convergence trick, net convergence is the first choice.
There is sequence $\{x_{n}\}_{n=1}^{\infty}$ on topological space X, and every pint of X is a accumulating point of the sequence.
X is uncountable set,the topology defined on X is $\mathcal{F}=\{X \backslash A |A \text{ is finite set of }X\}$, then for any sequence $\{x_{n}\}_{n=1}^{\infty}$ with $x_{n}\neq x_{m}$ when $n\neq m$,we can get every pint of X is a accumulating point.
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