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danflapjax committed Mar 27, 2024
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2 changes: 1 addition & 1 deletion theorems/T000044.md
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name: Counterexamples in Topology
---

In a {P183} space, every open set is itself closed.
In a {P185} space, every open set is itself closed.

Asserted on Figure 9 of {{doi:10.1007/978-1-4612-6290-9}}.
2 changes: 1 addition & 1 deletion theorems/T000469.md
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P000129: true
---

If a {P183} space is {P36}, then the partition generating it only has a single element, i.e. the space is {P129}.
If a {P185} space is {P36}, then the partition generating it only has a single element, i.e. the space is {P129}.
2 changes: 1 addition & 1 deletion theorems/T000470.md
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Expand Up @@ -6,4 +6,4 @@ then:
P000146: true
---

The partition generating a {P183} space is a refinement of any open cover.
The partition generating a {P185} space is a refinement of any open cover.
2 changes: 1 addition & 1 deletion theorems/T000471.md
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name: Topological group on Wikipedia
---

A the partition elements of a {P86} {P182} space must be all be the same cardinality, so it is a product of a {P52} space and an {P129} space.
A the partition elements of a {P86} {P185} space must be all be the same cardinality, so it is a product of a {P52} space and an {P129} space.
Given any nonzero cardinal, there is a group with that cardinality (for example, a cyclic group in the finite case and $\oplus_{k\in\kappa}\mathbb{Z}$ for infinite cardinal $\kappa$). When the group is given either the discrete or indiscrete topology, the group operations are continuous. Thus, the topological product of them admits a group structure from the product group of the two.

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