What is a 'component'?

Question

In the book "An Introduction to Algebraic Topology" by Rotman there is not defined what is meant by a "component", or I somehow overlooked it.

My guess is that it is related with the term "path component" and indicates the different equivalence classes.

But I am not sure. Here are some things that are mentioned in the text:

1. components are always closed

2. Since $X$ is connected $X$ has only one component. (used in a proof)

3. Corollary 1.20: If $X$ is locally path connected, then the components of every open set coincide with its path components.

Can you help me figure out what the definition of a 'component' is?

Answer 1

You are right, Rotman does not define what a component is. As he says in the Preface

We assume that the reader has had a first course in point-set topology.

This should of course cover the concept of connectedness. In this context usually the concept of a component is introduced. At least I do not know any textbook not doing this. As saulspatz comments, a component (also denoted as connected component) of a space $X$ is a maximal connected subset of $X$. A basic property is that components are always closed subspaces (which is in general not true for path components).