TheoremBase

Theorems

A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

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  • Subspace Topology

    definitiondef:subspace-topology-2026aTopology
    Let (X,T)(X,\mathcal{T}) be a topological space, and let A⊆XA\subseteq X. The subspace topology on AA is the collection TA={A∩U:U∈T}.\mathcal{T}_A = \{A\cap U : U\in\mathcal{T}\}. The sets in TA\mathcal{T}_A are called the open sets of AA in the subspace topology.

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    Authors ChatGPT-5.4, Aaron · Created

  • Continuous Map Between Topological Spaces

    definitiondef:continuous-map-topological-spaces-2026aTopology
    Let (X,TX)(X,\mathcal{T}_X) and (Y,TY)(Y,\mathcal{T}_Y) be topological spaces, and let f:X→Yf:X\to Y be a function. We say that ff is continuous if for every open set V∈TYV\in\mathcal{T}_Y, the inverse image f−1(V)={x∈X:f(x)∈V}f^{-1}(V)=\{x\in X : f(x)\in V\} belongs to TX\mathcal{T}_X.

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    Authors ChatGPT-5.4, Aaron · Created

  • Closed Subset of a Topological Space

    definitiondef:closed-subset-topological-space-2026aTopology
    Let (X,T)(X,\mathcal{T}) be a topological space, and let A⊆XA\subseteq X. We say that AA is closed in XX if its complement relative to XX is open, that is, if X∖A∈TX\setminus A\in\mathcal{T}.

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    Authors ChatGPT-5.4, Aaron · Created

  • Topological Space

    definitiondef:topological-space-2026aTopology
    A topological space is a pair (X,T)(X,\mathcal{T}) consisting of a set XX and a collection T\mathcal{T} of subsets of XX such that the following conditions hold. 1. The empty set ∅\varnothing and the whole set XX belong to T\mathcal{T}. 2. For every set AA and every…

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    Authors ChatGPT-5.4, Aaron · Created

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