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spaces/S000001/README.md

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- Finite Discrete Topology
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counterexamples_id: 1
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refs:
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- doi: 10.1007/978-1-4612-6290-9
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- zb: "0386.54001"
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name: Counterexamples in Topology
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- wikipedia: Discrete_space
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name: Discrete Space on Wikipedia
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---
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Let $X=2=\{0,1\}$ be the space containing two points and
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let all subsets of $X$ be open.
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Defined as counterexample #1 ("Finite Discrete Topology")
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in {{doi:10.1007/978-1-4612-6290-9}}.
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in {{zb:0386.54001}}.

spaces/S000001/properties/P000052.md

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space: S000001
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property: P000052
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value: true
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refs:
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- doi: 10.1007/978-1-4612-6290-9
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name: Counterexamples in Topology
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---
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All subsets of this space are open by definition.

spaces/S000002/README.md

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- wikipedia: Discrete_space
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name: Discrete Space on Wikipedia
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---
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Let $X=\omega=\{0,1,2,\dots\}$ be the space containing countably-many
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points and let all subsets of $X$ be open.
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spaces/S000002/properties/P000052.md

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space: S000002
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property: P000052
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value: true
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refs:
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- doi: 10.1007/978-1-4612-6290-9
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name: Counterexamples in Topology
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---
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All subsets of the space are open by definition.

spaces/S000002/properties/P000181.md

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space: S000002
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property: P000181
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value: true
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refs:
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- doi: 10.1007/978-1-4612-6290-9
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name: Counterexamples in Topology
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---
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Its cardinality is \(\aleph_0\) by definition.
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By definition.

spaces/S000003/README.md

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- Uncountable Discrete Topology
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counterexamples_id: 3
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refs:
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- doi: 10.1007/978-1-4612-6290-9
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- zb: "0386.54001"
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name: Counterexamples in Topology
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- wikipedia: Discrete_space
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name: Discrete Space on Wikipedia
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---
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Let $X=\mathbb R$ be the set of all real numbers,
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and let all subsets of $X$ be open.
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Defined as counterexample #3 ("Uncountable Discrete Topology")
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in {{doi:10.1007/978-1-4612-6290-9}}.
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in {{zb:0386.54001}}.

spaces/S000003/properties/P000052.md

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space: S000003
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property: P000052
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value: true
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refs:
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- doi: 10.1007/978-1-4612-6290-9
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name: Counterexamples in Topology
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---
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All subsets of the space are open by definition.

spaces/S000004/README.md

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name: Indiscrete Topology on a Two-Point Set
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counterexamples_id: 4
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refs:
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- doi: 10.1007/978-1-4612-6290-9
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- zb: "0386.54001"
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name: Counterexamples in Topology
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- wikipedia: Trivial_topology
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name: Trivial topology on Wikipedia
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---
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Let $X=2=\{0,1\}$ be the space containing two points and
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let only the sets $X$ and $\emptyset$ be open.
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Defined as counterexample #4 ("Indiscrete Topology")
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in {{doi:10.1007/978-1-4612-6290-9}}.
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in {{zb:0386.54001}}.

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