Let \( X \) be an uncountable set. Let the topology on \( X \) be defined by declaring a subset \( U \subset X \) to be open if \( X - U \) is either empty or finite or countable, and the empty set to be open. Then, which of the following is/are TRUE?
Show Hint
- Every compact subset of \( X \) is closed
- Every closed subset of \( X \) is compact
- \( X \) is \( T_1 \) (singleton subsets are closed) but not \( T_2 \) (Hausdorff)
- \( X \) is \( T_2 \) (Hausdorff)
The Correct Option is A, C
Solution and Explanation
The topology on \( X \) is defined such that a subset \( U \subset X \) is open if its complement \( X - U \) is either empty, finite, or countable. This is a specific topology called the cofinite topology. In this topology:
Every cofinite subset of \( X \) is open.
Singleton sets are closed because their complement is cofinite (i.e., countable).
A set is compact in this topology if every open cover has a finite subcover. Since the open sets are cofinite, compact sets are closed.
Step 2: Compact Subsets and Closedness (Option A)
In the cofinite topology, every compact subset of \( X \) is closed. This is because the complement of a compact set is cofinite, and the complement of a cofinite set is finite or countable, which is the condition for closed sets in this topology.
Step 3: Closed Subsets and Compactness (Option B)
Every closed subset of \( X \) is not necessarily compact. A closed set can be uncountable, and uncountable sets in this topology do not satisfy the compactness condition. Thus, Option B is not true.
Step 4: \( T_1 \) and \( T_2 \) Conditions (Option C and D)
The space is \( T_1 \) because singleton sets are closed. However, it is not Hausdorff because two distinct points cannot be separated by disjoint open sets in this topology. Thus, Option C is true, and Option D is false.
Step 5: Conclusion
Thus, the correct answers are \( \boxed{A} \) and \( \boxed{C} \).
Final Answer
\[ \boxed{A \quad \text{Every compact subset of } X \text{ is closed}} \] \[ \boxed{C \quad X \text{ is } T_1 \text{ (singleton subsets are closed) but not } T_2 \text{ (Hausdorff)}} \]
Top Questions on Subsets
- Let \( E \subset F \) and \( F \subset K \) be field extensions which are not algebraic. Let \( \alpha \in K \) be algebraic over \( F \) and \( \alpha \notin F \). Let \( L \) be the subfield of \( K \) generated over \( E \) by the coefficients of the monic polynomial of minimal degree over \( F \) which has \( \alpha \) as a zero. Then, which of the following is/are TRUE?
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Consider the following statements about such a ring \( R \):
P1: \( R \) is isomorphic to the product of two rings \( R_1 \) and \( R_2 \).
P2: \( \exists r_1, r_2 \in R \) such that \( r_1^2 = r_1 \neq 0 \), \( r_2^2 = 0 \), and \( r_1 + r_2 = 1 \).
P3: \( R \) has ideals \( I_1, I_2 \subset R \) with \( R \neq I_1 \), \( (0) \neq I_2 \), and \( R = I_1 + I_2 \) and \( I_1 \cap I_2 = (0) \).
P4: \( \exists a, b \in R \) with \( a \neq 0 \), \( b \neq 0 \) such that \( ab = 0 \).
Then, which of the following is/are TRUE? - In the following, all subsets of Euclidean spaces are considered with the respective subspace topologies. Define an equivalence relation \( \sim \) on the sphere \[ S = \left\{ (x_1, x_2, x_3) \in \mathbb{R}^3 : x_1^2 + x_2^2 + x_3^2 = 1 \right\} \] by \( (x_1, x_2, x_3) \sim (y_1, y_2, y_3) \) if \( x_3 = y_3 \), for \( (x_1, x_2, x_3), (y_1, y_2, y_3) \in S \). Let \( [x_1, x_2, x_3] \) denote the equivalence class of \( (x_1, x_2, x_3) \), and let \( X \) denote the set of all such equivalence classes. Let \( L : S \to X \) be given by \[ L\left( (x_1, x_2, x_3) \right) = [x_1, x_2, x_3]. \] If \( X \) is provided with the quotient topology induced by the map \( L \), then which one of the following is TRUE?
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S1: For every compact subset \( K \) of \( \mathbb{R} \), \( f^{-1}(K) \) is compact.
S2: For every compact subset \( K \) of \( \mathbb{R} \), \( g^{-1}(K) \) is compact. Then, which one of the following is correct?Consider the following subsets of the Euclidean space \( \mathbb{R}^4 \):
\( S = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 0 \} \),
\( T = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 1 \} \),
\( U = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = -1 \} \).
Then, which one of the following is TRUE?
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