326: Basis Determines Topology

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A description/proof of that basis determines topology

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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
• 3: Note

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Starting Context

• The reader knows a definition of topological space.
• The reader knows a definition of basis of topological space.
• The reader admits some criteria for any collection of open sets to be a basis.

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Target Context

• The reader will have a description and a proof of the proposition that any basis of any topological space determines the topology.

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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 basis, $\{B_{\alpha }\}$, of any topological space, $T$, determines the topology of $T$.

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2: Proof

For any open set, $U$, on $T$, around any point, $p\in U$, there is an open set, $U_p\subseteq U$, by the local criterion for openness. There is an open set, $B_{\alpha }\subseteq U_p$, in the basis, by the definition of basis. $U$ is the union of such open sets in the basis, by some criteria for any collection of open sets to be a basis.

Supposing that there are 2 topologies of the same set as the topological spaces, $T$ and $T^{\prime }$, with the same basis, any open set, $U$, on $T$ is the union of some open sets in the basis, so, $U$ is also an open set on $T^{\prime }$, and likewise, any open set, $U^{\prime }$, on $T^{\prime }$ is also an open set on $T$, so, $T=T^{\prime }$, which is what this proposition means.

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3: Note

So, the topological can be said to 'be generated by the basis'.

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References

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