Section 25*: Components and Local Connectedness

1. A component of $X$ is an equivalence class given by the equivalence relation: $x\sim y$ iff there is a connected subspace containing both.
2. A path component of $X$ is an equivalence class given by the equivalence relation: $x\sim y$ iff there is a path connecting them.
3. Facts about (path) components:
4. The (path) components of $X$ are (path) connected disjoint subspaces of $X$ whose union is $X$ such that each nonempty (path) connected subspace of $X$ intersects exactly one of them.
5. Each path component lies within a component.
6. Components are closed. If there are finitely many of them then they are open as well.
7. Local (path) connectedness. A space is locally (path) connected at a point if every neighborhood of the point contains a (path) connected sub-neighborhood. A space is locally (path) connected if it is locally (path) connected at every point.
8. A space $X$ is locally (path) connected iff every (path) component of every open set is open in $X$ .
9. If a space is locally path connected then the components and the path components are the same.
10. An open subset (as a subspace) of a locally (path) connected space is locally (path) connected.
11. A connected open subset of a locally path connected space is path connected.
12. Continuous functions and local connectedness:
13. The image of a locally connected space under a quotient map is locally connected.
14. A weakly locally connected at $x$ space $X$ is a space such that every neighborhood of $x$ contains a connected subspace of $X$ which contains a neighborhood of $x$ . $X$ is weakly locally connected if it is  weakly locally connected at every point.
15. Weak local connectedness of the whole space implies the local connectedness.
16. Weak local connectedness at a point does NOT imply the local connectedness at the point.
17. A quasi-component of $X$ is an equivalence class given by the equivalence relation: $x\sim y$ iff there is no separation of $X$ into two open sets each containing a point $x$ or $y$ .
18. A component is contained within a quasi-component, so that if the space is connected then there is only one quasi-component.
19. If a space is locally connected then the components and the quasi-components are the same.
20. The quasi-component of $X$ containing $x$ is the intersection of all subsets of $X$ containing $x$ that are open and closed at the same time.