Difference between revisions of "Connected (topology)"

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Revision as of 21:29, 30 September 2016

Definition

A topological space

$$(X,\mathcal{J})$$ is connected if there is no separation of $$X$$ ^[1] A separation of $X$ is:

• A pair of non-empty open sets in $X$, which we'll denote as $$U,\ V$$ where:
  1. $$U\cap V=\emptyset$$ and
  2. $$U\cup V=X$$

If there is no such separation then the space is connected^[2]

Equivalent definition

This definition is equivalent (true if and only if) the only empty sets that are both open in $X$ are:

1. $\emptyset$ and
2. $X$ itself.

I will prove this claim now:

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 $\emptyset$^[3]

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:
  • $$Y\subseteq U\cup V$$ and
  • $$U\cap V\subseteq C(Y)$$ and
  • Both $$U\cap Y\ne\emptyset$$ and $$V\cap Y\ne\emptyset$$

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

Results

References

1. Jump up ↑ Topology - James R. Munkres - 2nd edition
2. Jump up ↑ Analysis - Part 1: Elements - Krzysztof Maurin
3. Jump up ↑ Introduction to topology - Mendelson - third edition