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 . Its topology, say is the collection of subsets of which satisfies certain conditions. In the literature, each member of the collection is called an open set.

Regarding to topologies we have the following basic facts:

- Given two topologies and on , we say that is
*stronger*(or finer or richer) than if . - Given a sequence in , we say that
*converges*to in topology of if for any neighborhood of , there exists some large number such that for all . (Here by the neighborhood of we mean that there exists an open set of , i.e. is a member of the topology , such that .)

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 and on a topological space , we are interested in comparing and in terms of the identity map .

Lemma 1. The identity map iscontinuous if and only ifis stronger than .

*Proof*.

The proof is relatively easy. Indeed, if the map is continuous, then the preimage of any is also a member of which immediately implies that includes .

Having Lemma 1 in hand, we now try to compare topologies using norms.