The aim of this note is to derive some connections between topologies of normed spaces in terms of the equivalency of norms and the convergence of sequences.
Topological space and its topology: First, we start with a topological space, call 
Regarding to topologies we have the following basic facts:
- Given two topologies
and
on
.
- Given a sequence
in
converges to
in topology
of
such that
for all
. (Here by the neighborhood
of
.)
The key ingredient to compare topologies is to make use of the identity map. In the following part, we state a result which shall be used frequently in this note.
Topologies under the identity map: Given two topologies 
Lemma 1. The identity map
Proof.
The proof is relatively easy. Indeed, if the map
is continuous, then the preimage of any
is also a member of
Having Lemma 1 in hand, we now try to compare topologies using norms.
Ngô Quốc Anh 