In topology, a clopen set is a set that is both open and closed in a given topological space. The existence of clopen sets depends on the specific topology defined on the space.

Here are some examples of clopen subsets:

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1. Trivial Example in the Discrete Topology

• In a discrete topology, every subset of the space is open and closed.
• For example:
  • Let $X = \{1, 2, 3\}$.
  • The topology $\tau$ consists of all subsets of $X$:
    $\tau = \{\emptyset, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}, X\}$.
  • Here, every subset is clopen because it is both open (by definition of the discrete topology) and closed (its complement is also open).

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2. Clopen Sets in the Indiscrete (Trivial) Topology

• In the indiscrete topology, the only clopen sets are $\emptyset$ (the empty set) and $X$ (the entire space).
• For example:
  • Let $X = \{a, b\}$.
  • The topology $\tau$ is $\{\emptyset, X\}$.
  • Both $\emptyset$ and $X$ are clopen because:
    • They are open by definition, and
    • Their complements are also in $\tau$.

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3. Clopen Sets in $\mathbb{R}$ with the Usual Topology

• In the usual topology on the real numbers $\mathbb{R}$, the only clopen sets are:
  • The empty set $\emptyset$, and
  • The entire space $\mathbb{R}$.
• This result follows from the connectedness of $\mathbb{R}$, which means no nontrivial subset can be both open and closed.

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4. Clopen Sets in a Finite Topological Space

• In any finite topological space, clopen sets often arise naturally:
  • For example, let $X = \{a, b\}$ with the topology $\tau = \{\emptyset, \{a\}, X\}$.
  • Here, the subsets $\emptyset$, $X$, and $\{a\}$ are all clopen because their complements are also open.

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5. Clopen Sets in a Two-Point Space (Sierpiński Space)

• Consider $X = \{a, b\}$ with the topology $\tau = \{\emptyset, \{a\}, \{a, b\} \}$:
  • $\emptyset$ and $X$ are clopen.
  • $\{a\}$ is not clopen because its complement $\{b\}$ is not open.

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Summary Table of Examples

Space	Clopen Sets
Discrete Space	All subsets
Indiscrete Space	$\emptyset$ and $X$
$\mathbb{R}$ with usual topology	$\emptyset$ and $\mathbb{R}$
Finite Spaces	Varies based on the topology

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Would you like further details on any of these examples or more complex scenarios?

Related Questions:

1. Why are clopen sets significant in topology?
2. How can we prove that $\mathbb{R}$ has no nontrivial clopen sets?
3. What is the role of connectedness in determining clopen sets?
4. Can a set be clopen in one topology but not in another?
5. What are clopen sets in the context of metric spaces?

Tip: In a connected space, the only clopen sets are $\emptyset$ and the whole space. This property can help identify connected spaces.