Connected (topology)

Definition

A topological space $$(X,\mathcal{J})$$ is connected if there is no separation of $$X$$

Separation

This belongs on this page because a separation is only useful in this definition.

A separation of $$X$$ is a pair of two non-empty open sets $$U,V$$ where $$U\cap V=\emptyset$$ where $$U\cup V=X$$

Equivalent definition

We can also say: A topological space $$(X,\mathcal{J})$$ is connected if and only if the sets $$X,\emptyset$$ are the only two sets that are both open and closed.

[Expand]

Theorem: A topological space $$(X,\mathcal{J})$$ is connected if and only if the sets $$X,\emptyset$$ are the only two sets that are both open and closed.

---

Connected $$\implies$$ only sets both open and closed are $$X,\emptyset$$

Suppose $$X$$ is connected and there exists a set $$A$$ that is not empty and not all of $$X$$ which is both open and closed. Then as :this is closed, $$X-A$$ is open. Thus $$A,X-A$$ is a separation, contradicting that $$X$$ is connected.

Only sets both open and closed are $$X,\emptyset\implies$$ connected

---

TODO:

---

Connected subset

A subset $A$ of a Topological space $(X,\mathcal{J})$ is connected if (when considered with the Subspace topology) the only two Relatively open and Relatively closed (in A) sets are $A$ and $\emptyset$