(1) Let X = {1, 2, 3, 4}. Determine which of the following are topologies on X: (a) T = {{1}, {2}, {3}, {4}}. (b) T = {?, {1, 2, 3, 4}}. (c) T = {?, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}}. (d) T = {?, {1, 2}, {2, 3}, {1, 2, 3}, {1, 2, 3, 4}}. (e) T = {?, {2}, {1, 2}, {2, 3}, {1, 2, 3, 4}}. (2) List all of the topologies on X = {1, 2}. (3) Prove that the following set is NOT a topology on R: T = {?, R} ? {(0, n) : n ? N}.

Name: (1) Let X = 1, 2, 3, 4. Determine which of the following are topologies on X: (a) T = 1, 2, 3, 4. (b) T = ?, 1, 2, 3, 4. (c) T = ?, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4. (d) T = ?, 1, 2, 2, 3, 1, 2, 3, 1, 2, 3, 4. (e) T = ?, 2, 1, 2, 2, 3, 1, 2, 3, 4. (2) List all of the topologies on X = 1, 2. (3) Prove that the following set is NOT a topology on R: T = ?, R ? (0, n) : n ? N.
Uploaded: 2023-01-17T17:43:53-08:00
Duration: 3 min 3 s
Channel: Vincenzo Zaccaro
Description: (1) Let X = 1, 2, 3, 4. Determine which of the following are topologies on X: (a) T = 1, 2, 3, 4. (b) T = ?, 1, 2, 3, 4. (c) T = ?, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4. (d) T = ?, 1, 2, 2, 3, 1, 2, 3, 1, 2, 3, 4. (e) T = ?, 2, 1, 2, 2, 3, 1, 2, 3, 4. (2) List all of the topologies on X = 1, 2. (3) Prove that the following set is NOT a topology on R: T = ?, R ? (0, n) : n ? N.

          (1) Let X = {1, 2, 3, 4}. Determine which of the following are topologies on X: (a) T = {{1}, {2}, {3}, {4}}. (b) T = {?, {1, 2, 3, 4}}. (c) T = {?, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}}. (d) T = {?, {1, 2}, {2, 3}, {1, 2, 3}, {1, 2, 3, 4}}. (e) T = {?, {2}, {1, 2}, {2, 3}, {1, 2, 3, 4}}. (2) List all of the topologies on X = {1, 2}. (3) Prove that the following set is NOT a topology on R: T = {?, R} ? {(0, n) : n ? N}.

(1) Let X = 1, 2, 3, 4. Determine which of the following are topologies on X: (a) T = 1, 2, 3, 4. (b) T = ?, 1, 2, 3, 4. (c) T = ?, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4. (d) T = ?, 1, 2, 2, 3, 1, 2, 3, 1, 2, 3, 4. (e) T = ?, 2, 1, 2, 2, 3, 1, 2, 3, 4. (2) List all of the topologies on X = 1, 2. (3) Prove that the following set is NOT a topology on R: T = ?, R ? (0, n) : n ? N.

Added by Purificaci-N M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Vincenzo Zaccaro

01/17/2023

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A topology on a set \( X \) is a collection \( T \) of subsets of \( X \) such that: Show more…

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Let X = {1, 2, 3, 4}. Determine which of the following are topologies on X: (a) T = {{1}, {2}, {3}, {4}}. (b) T = {∅, {1, 2, 3, 4}}. (c) T = {∅, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}}. (d) T = {∅, {1, 2}, {2, 3}, {1, 2, 3}, {1, 2, 3, 4}}. (e) T = {∅, {2}, {1, 2}, {2, 3}, {1, 2, 3, 4}}. (2) List all of the topologies on X = {1, 2}. (3) Prove that the following set is NOT a topology on R: T = {∅, R} ∪ {(0, n) : n ∈ N}.