214: Topological Space Is Connected iff Its Open and Closed Subsets Are Only It and Empty Set

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A description/proof of that topological space is connected iff its open and closed subsets are only it and empty set

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Topics

About: topological space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

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Starting Context

• The reader knows a definition of connected topological space.
• The reader knows a definition of closed set.

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Target Context

• The reader will have a description and a proof of the proposition that any topological space is connected if and only if its open and closed subsets are only the topological space and the empty set.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Description

Any topological space, $T$, is connected if and only if its open and closed subsets are only $T$ and $\mathrm{\varnothing }$.

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2: Proof

Suppose that the open and close subsets of $T$ are only $T$ and $\mathrm{\varnothing }$. Suppose that $T$ was not connected. $T=U_1\cup U_2$, $U_1\cap U_2=\mathrm{\varnothing }$ where $U_i$ would be a non-empty open set on $T$. $T\setminus U_1=U_2$ would be closed, so, $U_2$ would be an open and closed subset, a contradiction, so, $T$ is connected.

Suppose that $T$ is connected. $T$ and $\mathrm{\varnothing }$ are open and closed subsets. Suppose that there was another open and closed subset, $U\subseteq T$. $T\setminus U$ would be open and non-empty, $U\cap (T\setminus U)=\mathrm{\varnothing }$, and $T=U\cup (T\setminus U)$, so, $T$ would not be connected, a contradiction, so, $T$ and $\mathrm{\varnothing }$ are the only open and closed subsets.

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References

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