Let M be a subspace of R where \(M= [1,2]\cup [3,4]\) then \([1,2]\) is


A) open in M

B) closed in M

C) both open and closed in M

D) neither open nor closed in M

Correct Answer:
C) both open and closed in M

Description for Correct answer:
Clearly [1, 2] is closed in M = [1, 3]\(\cup\) [3, 4]

\(\textbf{Result:}\)

Let M be a metric space and \(M_{1}\) is a subspace of M.

Let \(A_{1}\subseteq M_{1}\). Then \(A_{1}\) is open in \(M_{1}\) if and only if there exists an open set A of M that \(A_{1}=A\cup M_{1}\)

Let \(\Large A= \left(\frac{1}{2},\frac{5}{2}\right) \)

and let \(A_{1}=[1, 2]\) then clearly

\(\Large A_{1}=[1,2]= \left(\frac{1}{2},\frac{5}{2}\right)\cap M \)

\(\therefore [1,2]\) is open in M.

Therefore \([1,2]\) is both open and closed in M.

Part of solved Real Analysis questions and answers : >> Elementary Mathematics >> Real Analysis








Comments

No comments available




Similar Questions
1). It R be the metric space then,
A). \(\left( 0,\frac{1}{2} \right]\) is not Open in [0, 2]
B). \(\left( 0,\frac{1}{2} \right]\) is open in [0, 2]
C). \(\left( 0,\frac{1}{2} \right]\) is closed in [0, 2]
D). None of these
-- View Answer
2). Z is
A). open in R
B). closed in R
C). bounded in R
D). none of these
-- View Answer
3). Every subset of a discrete metric space
A). only open
B). only closed
C). both open and closed
D). neither open nor closed
-- View Answer
4). In with usual metric, every singleton set is
A). open
B). closed
C). both open and closed
D). none of these
-- View Answer
5). In any metric space M, \(\phi\) and M are
A). open
B). closed
C). both open and closed
D). neither open nor closed
-- View Answer


6). Any finite suliset of a metric space is
A). open
B). closed
C). both open and closed
D). neither open nor closed
-- View Answer
7). A subset M of \(R^{2}\) is compact if and only if M is ____.
A). closed
B). bounded
C). closed and bounded
D). none of these
-- View Answer
8). Let (M, d) be the discrete metric space and A be any subset of A. Then the derived set of A (the set of all limit points of A) is,
A). A
B). M
C). \(\phi\)
D). \(A^{c}\)
-- View Answer
9). Any finite subset of a metric space has
A). limit points
B). no limit points
C). both (A) and (B) are true
D). none of these
-- View Answer
10). Examine the convergence of \(\Large\int\limits_{0}^{1}\frac{dx}{x^{2}}\)
A). convergent
B). divergent
C). converges to 1
D). converges to 0
-- View Answer