List of top Mathematics Questions on mathematical reasoning asked in GATE MA

Let \( X \) be the space \( {R}/{Z} \) with the quotient topology induced from the usual topology on \( {R} \). Consider the following statements:
1. \( X \) is compact.
2. \( X \setminus \{z\} \) is connected for any \( z \in X \).
Which one of the following is correct?

Consider the following statements:
1. Every compact Hausdorff space is normal.
2. Every metric space is normal.
Which one of the following is correct?

Consider the topology on \( {Z} \) with basis \( S(a,b) = \{an + b : n \in {Z\} \), where \( a, b \in {Z} \) and \( a \neq 0 \). Consider the following statements:} 1. \( S(a, b) \) is both open and closed for each \( a, b \in {Z} \) with \( a \neq 0 \).
2. The only connected set containing \( x \in {Z} \) is \( \{x\} \).
Which one of the following is correct?

Let \( X \) be a topological space and \( A \subseteq X \). Given a subset \( S \) of \( X \), let \( {int}(S), \partial S, \) and \( \overline{S} \) denote the interior, boundary, and closure, respectively, of the set \( S \). Which one of the following is NOT necessarily true?