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Brief Review of Structures and Spaces in Mathematics

Junsheng Wang

August 15, 2023

1 Introduction

A math major would encounter many kinds of ”spaces” and ”structures” during their

study. To the writer’s knowledge, A space in mathematics is a set with a speciﬁc struc-

ture. Several modules, for example, abstract algebra, topology, and functional analysis

will talk about some classic spaces. However, it may cause confusion, at least the writer

is confused, that what is the relationship between all these spaces. To clear the mist,

this review will ﬁrst introduce the deﬁnition of soem spaces in college mathematics and

then demonstrate how they are related to each other.

2 Deﬁnition of Common Spaces

2.1 Topological Space

T is a collection of subsets of a set X , T is called a topology on X if:

(1)∅ and X are in T .

(2)the union of any sub-collection of T is in T .

(3)the intersection of any ﬁnite sub-collection of T is in T .

A set X which is assigned a topology is called a topological space, denoted by (X , T ).

(X , T ) is called a metrizable space if the topology T is induced by some metric d

on X . Metric space is a metrizable space along with the metric d which induces the

topology of X .

2.2 Normed Linear Space

A real or complex linear space X is called normed linear space if there exists a function

∥x∥ from X to R

+

satisﬁes:

(1)∥x∥ ≥ 0 and ∥x∥ = 0 ⇐⇒ x = 0

(2)∀α ∈ R(orC), ∀x ∈ X , ∥αx∥ = |α|∥x∥

(3)∀x, y ∈ X , ∥x + y∥ ≤ ∥x∥ + ∥y∥

1

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