Topological space Study Guide

📖 Core Concepts Topological space – a set \(X\) together with a collection \(\mathcal T\) of subsets (the open sets) satisfying the three topology axioms. Open / Closed – \(U\subseteq X\) is open if \(U\in\mathcal T\); \(C\subseteq X\) is closed iff its complement \(X\setminus C\) is open. Neighbourhood – any set \(U\) that contains an open set around a point \(x\); the family \(\mathcal N(x)\) must be non‑empty for every \(x\). Finer vs. coarser topology – \(\mathcal T1\) is finer than \(\mathcal T2\) if \(\mathcal T2\subseteq\mathcal T1\). Continuity – \(f:X\to Y\) is continuous ⇔ for every open \(O\subseteq Y\), \(f^{-1}(O)\) is open in \(X\). Homeomorphism – a bijective continuous map whose inverse is also continuous; it captures “same shape’’ up to topology. Metric‑induced topology – open balls \(Br(x)=\{y\mid d(x,y)<r\}\) generate the topology. Initial / final topology – the coarsest (resp. finest) topology making a given family of maps continuous. --- 📌 Must Remember Axioms of a topology: \(\emptyset, X\in\mathcal T\). Arbitrary unions of members of \(\mathcal T\) are in \(\mathcal T\). Finite intersections of members of \(\mathcal T\) are in \(\mathcal T\). Continuous ⇔ preimage of every open set is open (equivalently, preimage of every closed set is closed). Subspace topology: open sets in \(Y\subseteq X\) are \(U\cap Y\) with \(U\) open in \(X\). Product topology (finite): basis = all products \(\prodi Ui\) where each \(Ui\) is open in \(Xi\). Product topology (infinite): basis = products \(\prodi Ui\) where all but finitely many \(Ui = Xi\). Quotient topology: \(V\subseteq Y\) is open iff \(q^{-1}(V)\) is open in \(X\) for a surjection \(q:X\to Y\). Cofinite topology on infinite \(X\): open sets = \(\emptyset\) + all subsets with finite complement. Lower limit (Sorgenfrey) topology on \(\mathbb R\): basis = half‑open intervals \([a,b)\). Order topology: generated by open intervals \((a,b)\) and rays \([a0,b), (a,b0]\). Topological invariant: property preserved under homeomorphisms (e.g., connectedness, compactness, \(T1\)). --- 🔄 Key Processes Verifying a topology on a set \(X\): Check \(\emptyset, X\) are included. Test arbitrary unions → stay in the collection. Test finite intersections → stay in the collection. Proving continuity of \(f:X\to Y\): Choose an arbitrary open \(O\subseteq Y\). Show \(f^{-1}(O)\) fits the description of an open set in \(X\) (often via subbasis, basis, or neighbourhood definition). Constructing the product topology: Write down projection maps \(\pii:\prod Xi\to Xi\). Declare the coarsest topology making each \(\pii\) continuous → generated by \(\pii^{-1}(Ui)\) for open \(Ui\subseteq Xi\). Forming a quotient topology: Start with surjection \(q:X\to Y\). Declare \(V\subseteq Y\) open iff \(q^{-1}(V)\) is open in \(X\). Using invariants to disprove homeomorphism: List a property (e.g., compactness). Show space \(A\) has it, space \(B\) does not → they cannot be homeomorphic. --- 🔍 Key Comparisons Finer vs. coarser Finer: more open sets → stronger topology (e.g., Sorgenfrey line finer than Euclidean). Coarser: fewer open sets → weaker topology (e.g., cofinite topology is coarser than discrete). Subspace vs. product topology Subspace: inherits openness by intersecting with a larger space. Product: built from projections; basis elements are products of opens, not intersections. Metric‑induced vs. arbitrary topology Metric: open balls give a countable basis, automatically Hausdorff. Arbitrary: may lack countable bases, may fail Hausdorff (e.g., cofinite on uncountable set). Initial vs. final topology Initial: coarsest making given maps from \(X\) continuous. Final: finest making given maps into \(Y\) continuous. --- ⚠️ Common Misunderstandings “Every topology comes from a metric.” False; many (e.g., cofinite, lower limit) are non‑metrizable. “Continuity = sequential continuity.” Only true in first‑countable spaces; in general nets are needed. “If a map is continuous, the image of an open set is open.” Wrong; continuity guarantees preimages of opens are open, not images. “Finer topology always makes more functions continuous.” It makes more maps into the space continuous, but fewer maps out of the space. --- 🧠 Mental Models / Intuition Topology as “lens” on a set – the open sets decide what it means for points to be “close” without measuring distance. Finer = clearer view – you can see more “small” neighborhoods; coarser = blurrier, only large chunks are open. Continuity = “no tearing” – think of pulling a rubber sheet: the preimage of any open region on the target sheet must already be an open region on the source sheet. Quotient = “gluing together” – identify points via \(q\); the topology on the quotient is the most permissive that still respects the glued structure. --- 🚩 Exceptions & Edge Cases Cofinite topology is \(T1\) but not Hausdorff (cannot separate two points by disjoint opens). Sorgenfrey line is normal but not metrizable; product \(\mathbb{R}\ell \times \mathbb{R}\ell\) is not normal. Infinite product topology: basis elements require all but finitely many coordinates to be the whole space; dropping this condition gives the box topology (strictly finer, often pathological). Initial topology may be trivial if the family of maps is empty (only \(\emptyset, X\) are open). --- 📍 When to Use Which Prove a set is a topological space → check the three axioms directly (quickest). Show a map is continuous → use the preimage‑of‑open‑sets criterion; if the codomain has a basis, it suffices to check preimages of basis elements. Construct a topology on a subset → always take the subspace topology (intersect with opens of the ambient space). Combine several spaces → use the product topology (finite: full product basis; infinite: finite‑support basis). Identify a space obtained by identification → apply the quotient topology definition. Compare two topologies on the same set → test inclusion of open families to decide finer/coarser. --- 👀 Patterns to Recognize “Every open set contains a basis element” – whenever a topology is described via a basis, look for the smallest open neighborhoods that generate all others. “Closed under arbitrary unions, finite intersections” – if a candidate collection fails either, it cannot be a topology. “Continuity ⇔ preimage of basis elements is open” – in exam questions, checking a basis is often enough. “Quotient topology = finest making the projection continuous” – remember the word finest; any larger topology would break continuity of the defining map. “Metric topology = open balls” – whenever a metric is given, the topology is automatically the one generated by those balls. --- 🗂️ Exam Traps Distractor: “If \(f\) is continuous, then \(f(U)\) is open for every open \(U\).” – This confuses the preimage condition with the image condition. Trap: Assuming the product of two non‑Hausdorff spaces must be non‑Hausdorff. Counterexample: product of two indiscrete spaces is indiscrete (still non‑Hausdorff), but product of a Hausdorff with any space is Hausdorff only if the other is Hausdorff. Misreading “finer” – Some students pick the smaller collection; remember “finer = more opens”. Edge‑case answer: “The lower limit topology on \(\mathbb R\) is metrizable because it has a basis of intervals.” – False; it is not generated by any metric. Over‑generalizing: “All topological invariants are numeric.” – Invariants include qualitative properties (connectedness, compactness). ---