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.

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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.

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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

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TODO:

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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 $\emptyset$

Useful lemma

Given a topological subspace $Y$ of a space $(X,\mathcal{J})$ we say that $Y$ is disconnected if and only if:

• $$\exists U,V\in\mathcal{J}$$ such that:
  • $$A\subseteq U\cup V$$ and
  • $$U\cap V\subseteq C(A)$$ and
  • Both $$U\cap A\ne\emptyset$$ and $$V\cap A\ne\emptyset$$

This is definition basically says there has to be a separation of $A$ that isn't just $A$ and the $\emptyset$ for $A$ to be disconnected, but the sets may overlap outside of A

Results

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Theorem:Given a topological subspace $Y$ of a space $(X,\mathcal{J})$ we say that $Y$ is disconnected if and only if $$\exists U,V\in\mathcal{J}$$ such that: $$A\subseteq U\cup V$$ , $$U\cap V\subseteq C(A)$$ , $$U\cap A\ne\emptyset$$ and $$V\cap A\ne\emptyset$$

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TODO: Mendelson p115

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Theorem: The image of a connected set is connected under a continuous map

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TODO: Mendelson p116

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