Basis and Subspace Topology MCQs

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Topic Notes: Basis and Subspace Topology

Quick Revision 3 min read 16 key points CSS, PMS, FPSC, PPSC

Section 1 of 4

Section:

1 1. Definition of a Basis

4 points

Definition

Basis for a Topology: A collection B of subsets of X is a basis for a topology T if for every x in X, there exists at least one B in B such that x is in B.
Theory

Basis Intersection Property: For any B1, B2 in B and x in B1 intersection B2, there must exist a B3 in B such that x is in B3 and B3 is a subset of B1 intersection B2.
Definition

Generated Topology: A set U is open in T if for every x in U, there exists a basis element B in B such that x is in B and B is a subset of U.
Fact

Basis Uniqueness: Different bases can generate the same topology. Two bases B and B' generate the same topology if every element of B is open in the topology generated by B', and vice versa.

2 2. Subspace Topology

4 points

Definition

Subspace Topology: If Y is a subset of X, the subspace topology on Y is defined as Ty = {U intersection Y | U is open in X}.
Theory

Subspace Basis: If B is a basis for X, then B_Y = {B intersection Y | B is in B} is a basis for the subspace topology on Y.
Fact

Openness in Subspace: A set U is open in Y if and only if U = V intersection Y for some open set V in X. Note: U is not necessarily open in X.
Theory

Transitivity of Subspace: If Z is a subspace of Y and Y is a subspace of X, then the subspace topology on Z inherited from Y is the same as that inherited from X.

3 3. Common Examples

4 points

Fact

Standard Topology on R: The collection of all open intervals (a, b) forms a basis for the standard topology on the real line R.
Fact

Lower Limit Topology (R_l): The collection of all half-open intervals [a, b) forms a basis for the lower limit topology on R, which is strictly finer than the standard topology.
Fact

Discrete Topology: If X has the discrete topology, then every subset Y of X inherits the discrete topology, where every singleton {y} is open in Y.
Fact

Ray Topology: The collection of all open rays (a, infinity) forms a basis for a topology on R, but it is strictly coarser than the standard topology.

4 4. Comparison and Exam Traps

4 points

Exam Trap

Openness Confusion: A set U open in subspace Y is not automatically open in the parent space X. Example: (0, 1] is open in the subspace [0, 1] of R, but not open in R.
Comparison

Basis vs. Topology: A basis is a collection of sets, while a topology is the collection of all unions of those basis elements. Not every collection of sets is a basis.
Exam Trap

Closed Sets in Subspace: If A is closed in Y and Y is closed in X, then A is closed in X. However, if Y is not closed in X, A may not be closed in X.
Comparison

Finer vs. Coarser: A topology T' is finer than T if T is a subset of T'. A basis B generates a finer topology if it contains more sets or larger sets than the basis for T.

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If X is a topological space and Y is a subspace, which property is inherited by Y from X?

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Given the set X = {a, b, c}, which of the following is a basis for a topology on X?

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Which of the following is true regarding the basis of a topology?

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If Y is a subspace of X, and U is a subset of Y, U is closed in Y if and only if:

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What is the smallest topology containing a given collection of sets S?

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If X has the discrete topology, what is the subspace topology on any subset Y of X?

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Which topology is generated by the basis consisting of all open rays (a, infinity)?

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In the subspace topology, if A is closed in Y and Y is closed in X, what can be said about A in X?

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If B is a basis for a topology T, what property must the intersection of two basis elements B1 and B2 satisfy?

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A For each x in B1 intersection B2, there exists B3 in B such that x is in B3 and B3 is a subset of B1 intersection B2 B The intersection must be empty C The intersection must be an element of B D The intersection must be a singleton set

Which of the following is a basis for the lower limit topology on R?

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Pedagogical Syllabus Blueprint

Basis and Subspace Topology MCQs: Preparation Strategy & Exam Blueprint

Examination Weightage & Commission Distribution

In Pakistan's premier competitive examinations—including the Federal Public Service Commission (FPSC General Recruitment & CSS MPT), Punjab Public Service Commission (PPSC Lecturer, Tehsildar & PMS Screening), KPPSC, SPSC, and BPSC—questions derived from Basis and Subspace Topology MCQs consistently constitute a decisive 10% to 25% weightage of the Topology paper. Testing agencies frequently evaluate both foundational factual recall and deeper analytical comprehension, requiring aspirants to master key dates, statutory articles, constitutional clauses, and thematic definitions.

High-Yield Conceptual Pillars & Distractor Analysis

Historical trend analysis across past five-year examination cycles demonstrates that examiner traps frequently involve closely matched numerical alternatives, chronologically contiguous treaties, and semantic subtleties in statutory phraseology. When preparing Basis and Subspace Topology MCQs, candidates must deliberately review the underlying rationale provided for every question rather than merely memorizing raw answer options. Understanding why incorrect distractors fail is the most reliable predictor of achieving top percentiles in computerized and optical mark recognition (OMR) screening tests.

Negative Marking Mitigation & Time Management Protocol

For examinations enforcing negative marking penalties (such as PPSC's 0.25-mark deduction per erroneous response or NTS gatekeeper quotas), an aggressive elimination technique must be deployed. Candidates should systematically rule out improbable options first to elevate probability thresholds from 25% to 50% or 100%. Allocating an average of 45 to 60 seconds per item during mock trials builds the psychological stamina and cognitive precision required to complete 100-question papers within official 90-minute time limits.

Recommended Reference Sources & Statutory Reading

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Solved Questions

High-Yield Exam Revision Cheat Sheet & Mnemonics

Topic Notes: Basis and Subspace Topology

1 1. Definition of a Basis

2 2. Subspace Topology

3 3. Common Examples

4 4. Comparison and Exam Traps

Basis and Subspace Topology MCQs: Preparation Strategy & Exam Blueprint

Examination Weightage & Commission Distribution

High-Yield Conceptual Pillars & Distractor Analysis

Negative Marking Mitigation & Time Management Protocol

Recommended Reference Sources & Statutory Reading

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