Exam focus: Compare topologies (finer/coarser) and know how the product topology differs from the box topology on infinite products — that difference is examined directly.
Introduction — Chapter 1, Lecture 1
NPTEL · Topology
Examples of Topological Spaces — Chapter 1, Lecture 2
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The standard example zoo: discrete, indiscrete, cofinite.
Product Topology — Chapter 2, Lecture 6
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Box and Product Topologies — Chapter 2, Lecture 8
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Exactly where box and product differ on infinite products.
Solution
(1) X × Y is a union of connected crosses (X × {y}) ∪ ({x} × Y) through a common point. (2) fails: identifying x ~ y in whenever yields an indiscrete-like quotient that is far from Hausdorff — quotients can destroy separation. (3) The separating open sets restrict to the subspace. (4) Open boxes and open discs generate each other.
Solution
Any open cover has a member missing only finitely many points; cover those with finitely many more sets. Cofinite is not Hausdorff (any two non-empty open sets meet).
Exam focus: Know the implication chain T₄ ⇒ T₃ ⇒ T₂ ⇒ T₁ ⇒ T₀ (with the regularity/normality convention) and which properties are hereditary or productive.
Continuity and Related Concepts — Chapter 1, Lecture 5
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T1 – Spaces and Hausdorff Spaces — Chapter 1, Lecture 4
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Separation Axioms — Chapter 5, Lecture 19
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Regular and Normal Spaces — Chapter 5, Lecture 20
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Normality is the badly-behaved axiom — note which constructions preserve it.
Solution
makes every singleton closed; in a finite space every subset is then a finite union of closed singletons, hence closed — so every subset is also open. That is the discrete topology, which for more than one point is maximally disconnected.
Exam focus: Heine–Borel is ℝⁿ-only. In general spaces use open covers or the finite intersection property; in metric spaces sequential compactness is equivalent.
Compact Spaces — Chapter 4, Lecture 15
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Properties of Compact Spaces — Chapter 4, Lecture 16
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Compact ⊆ Hausdorff ⇒ closed, and continuous images stay compact.
Solution
Only option C is closed and bounded.
Solution
(1) Closed subsets of the compact domain are compact, their images are compact and hence closed in the Hausdorff codomain — so the inverse map is continuous. (2) A closed subset inherits the finite subcover property. (3) needs Hausdorff: in the cofinite topology on , every subset is compact, including non-closed ones. (4) is Tychonoff's theorem.
Exam focus: Components are always closed but not always open; in ℚ every component is a single point. Use clopen sets or a continuous surjection onto {0,1} to disprove connectedness.
Connected Spaces — Chapter 3, Lecture 10
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Pathwise Connected Spaces — Chapter 3, Lecture 13
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Path-connected ⇒ connected; the converse is the topologist's sine curve.
Components and Introduction to Compact Spaces — Chapter 4, Lecture 14
NPTEL · Topology
Solution
(a) disconnected (split at any rational). (b) two components (det > 0, det < 0). (c) connected but not path-connected. (d) two branches.
Solution
(1) A disconnection of the image pulls back to one of the domain. (2) Any set squeezed between a connected set and its closure is connected. (3) A disconnection would disconnect the interval [0, 1] along some path. (4) fails for the topologist's sine curve {(x, sin(1/x)) : 0 < x ≤ 1} ∪ ({0} × [−1, 1]): connected as the closure of a connected graph, but no path reaches the vertical segment.
Exam focus: These four spaces are the exam's stock counterexamples — memorise the property table and you can answer most 'which of the following is true' topology items by elimination.
Examples of Topological Spaces — Chapter 1, Lecture 2
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The counterexample zoo: discrete, indiscrete and cofinite topologies.
Interior Points, Limit Points — Chapter 1, Lecture 3
NPTEL · Topology
How closure and interior behave in these non-metric spaces.
Solution
It is in bijection with {0,1}, hence uncountable, and the removed intervals have total length 1.
Solution
Any two non-empty open sets meet (their complements are finite), so X is not Hausdorff but is connected; the closure of an infinite set is X.
Solution
(1) Given an open cover, pick any non-empty member U; its complement is finite, so finitely many further members cover the rest — X is compact. (2) Any two non-empty open sets have cofinite complements and therefore intersect, so points cannot be separated: X is not Hausdorff. (3) For the same reason two non-empty open sets always meet, so X cannot be split into disjoint non-empty open sets — X is connected. (4) The closed sets are exactly the finite sets together with X, so an infinite proper subset such as is not closed.
Solution
[0, 1) is a basis element, hence open. Its complement is a union of basis elements [−n, 0) and [1, n), hence open too — so [0, 1) is clopen. The abundance of clopen sets is why the Sorgenfrey line is totally disconnected.