Relative openness differs from ambient openness

Goal and scope

Compute the subspace topology and all relative closed sets of Y={b,c}.

A complete finite ambient topology; relative complements are inside Y.

Givens and symbols

• sets
• topology
• definition_here

• G1 — X={a,b,c}, tau_X={empty,{a},{a,b},X}, Y={b,c}.

• Y — subspace carrier
• \tau_Y — relative topology

Rules and sources

• mko_definition · cited — The induced topology is {U∩Y:U∈tau_X}, normalizing duplicate and empty intersections. Relative opens need not be ambient open and their witnesses U need not lie inside Y. Relative closedness uses open complements inside Y, or intersections with ambient closed sets. Y is the relative universe in every topology obligation. · mko-subspace-topology
• sets · prerequisite — Compute unions, intersections, complements, inclusion and preimages element by element.
• topology · prerequisite — A topology contains empty and whole sets and is closed under arbitrary unions and finite intersections; a finite family admits exhaustive checks.
• definition_here · prerequisite — The subspace topology consists of intersections of ambient opens with the subspace carrier; relative closed sets are complements of relative opens inside that carrier.

Main derivation

\varnothing\subset\{a\}\subset\{a,b\}\subset X

\tau_X\text{ is a topology},\quad Y\subseteq X

\varnothing\cap Y,\ \{a\}\cap Y

\varnothing,\ \varnothing

\{a,b\}\cap Y,\ X\cap Y

\{b\},\ Y

\tau_Y=\{U\cap Y:U\in\tau_X\}

\tau_Y=\{\varnothing,\{b\},Y\}

\varnothing\subset\{b\}\subset Y

\tau_Y\text{ is a topology on }Y

\{Y\setminus V:V\in\tau_Y\}

\mathcal F_Y=\{Y,\{c\},\varnothing\}

\{b\}\in\tau_Y

\{b\}\notin\tau_X

Y\setminus\{c\}=\{b\}\in\tau_Y

\{c\}\text{ is relatively closed};\quad\{b\}\text{ is not relatively closed}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · S06, S07 — Relative closedness is checked against the entire relative topology.
• S07 · premise_audit · G1, S04 — The ambient-open witness is {a,b}, not {b} itself.
• S06 · premise_audit · S04 — Complementing in X instead could produce sets not contained in Y.
• S05 · premise_audit · S04 — The whole-space member is Y, not X.
• S04 · premise_audit · S02, S03 — All four ambient opens were intersected, so no relative open is missing.
• S03 · inverse_equivalence · G1 — The witnessing ambient open {a,b} need not lie inside Y.
• S02 · inverse_equivalence · G1 — An empty intersection is still a legal relative open.
• S01 · premise_audit · G1 — Parent validity is not inferred circularly from the desired subspace family.

Error controls and negative controls

List every intersection

Traverse every ambient open before deduplicating.

Relative universe

Relative complements and axioms consistently use Y.

• The witnessing ambient open must lie inside Y. → reject — The witness {a,b} for {b} is not contained in Y.
• {b} is open in X. → reject — It is absent from the complete ambient topology.
• The relative whole space is still X. → reject — The relative topology has whole space Y.

Result and limitations

tau_Y={empty,{b},Y}; the relative closed sets are {empty,{c},Y}.

Verification and source binding

0 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 24419aa4e704355f75ed3b708e05ec2f2252aa18cea0d39bb65e43fd886167b6
• original source SHA-256: 24419aa4e704355f75ed3b708e05ec2f2252aa18cea0d39bb65e43fd886167b6
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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