Q1: A collection $\tau$ of subsets of a set $X$ is a topology on $X$ if it satisfies which condition?
A: Contains $\emptyset$ and $X$
B: Closed under arbitrary unions
C: Closed under finite intersections
D: All of the above
Correct Answer: D
Q2: In the discrete topology on $X$, every subset is:
A: Only open
B: Only closed
C: Both open and closed
D: Neither open nor closed
Correct Answer: C
Q3: The indiscrete topology on $X$ contains exactly:
A: The power set of $X$
B: $\emptyset$ and $X$
C: Only singleton sets
D: Finite subsets only
Correct Answer: B
Q4: If $\tau_1 \subset \tau_2$, we say $\tau_1$ is ______ than $\tau_2$.
A: Finer
B: Coarser
C: Stronger
D: Equivalent
Correct Answer: B
Q5: In the cofinite topology on an infinite set $X$, a set $A$ is open if $X \setminus A$ is:
A: Countable
B: Finite
C: Infinite
D: Empty
Correct Answer: B
Q6: The intersection of any collection of topologies on $X$ is:
A: Always a topology
B: Never a topology
C: A topology only if the collection is finite
D: A topology only if $X$ is finite
Correct Answer: A
Q7: The closure of a set $A$, denoted $\overline{A}$, is the intersection of all ______ sets containing $A$.
A: Open
B: Closed
C: Dense
D: Bounded
Correct Answer: B
Q8: A point $x$ is a limit point of $A$ if every neighborhood of $x$ contains at least one point of $A$ ______ $x$.
A: Including
B: Other than
C: Exactly like
D: Equal to
Correct Answer: B
Q9: The interior of $A$ is the ______ open set contained in $A$.
A: Smallest
B: Largest
C: Only
D: Complementary
Correct Answer: B
Q10: A set $A$ is dense in $X$ if:
A: $\text{Int}(A) = X$
B: $\overline{A} = X$
C: $\text{Bd}(A) = X$
D: $A = X$
Correct Answer: B
Q11: The set of rational numbers $\mathbb{Q}$ is ______ in $\mathbb{R}$ with the usual topology.
A: Closed
B: Open
C: Dense
D: Discrete
Correct Answer: C
Q12: A collection $\mathcal{B}$ is a base for $\tau$ if every open set is a ______ of elements of $\mathcal{B}$.
A: Finite intersection
B: Arbitrary union
C: Complement
D: Finite union
Correct Answer: B
Q13: A collection $\mathcal{S}$ is a sub-base if finite ______ of its elements form a base.
A: Unions
B: Intersections
C: Complements
D: Differences
Correct Answer: B
Q14: A space $X$ is called Second Countable if it has a ______ base.
A: Finite
B: Countable
C: Uncountable
D: Empty
Correct Answer: B
Q15: The boundary of $A$ is defined as $\text{Bd}(A) = $:
A: $\overline{A} \setminus \text{Int}(A)$
B: $\overline{A} \cap \text{Int}(A)$
C: $\text{Int}(A) \setminus \overline{A}$
D: $\overline{A} \cup \text{Int}(A)$
Correct Answer: A
Q16: A map $f: X \to Y$ is continuous if for every open set $V \subset Y$, $f^{-1}(V)$ is ______ in $X$.
A: Closed
B: Open
C: Compact
D: Empty
Correct Answer: B
Q17: A function $f$ is a homeomorphism if it is a continuous bijection and its ______ is also continuous.
A: Derivative
B: Inverse
C: Adjoint
D: Square
Correct Answer: B
Q18: If $f$ is a homeomorphism, then $X$ and $Y$ are called ______ spaces.
A: Isometric
B: Topologically equivalent
C: Homomorphic
D: Isomorphic
Correct Answer: B
Q19: A map $f$ is called an 'open map' if the image of every open set is ______.
A: Closed
B: Open
C: Compact
D: Connected
Correct Answer: B
Q20: An isometry between metric spaces preserves:
A: Area
B: Volume
C: Distance
D: Continuity only
Correct Answer: C
Q21: Every isometry is a ______ map.
A: Homeomorphic
B: Constant
C: Discontinuous
D: Linear only
Correct Answer: A
Q22: Uniform continuity is a concept that requires a ______ structure.
A: Topological only
B: Metric (or Uniform)
C: Algebraic
D: Discrete
Correct Answer: B
Q23: Which of the following is a topological property (invariant under homeomorphism)?
A: Boundedness
B: Completeness
C: Compactness
D: Specific Diameter
Correct Answer: C
Q24: In $\mathbb{R}$ with the usual topology, the interior of the set $[0, 1]$ is:
A: $[0, 1]$
B: $(0, 1)$
C: $\{0, 1\}$
D: $\emptyset$
Correct Answer: B
Q25: The identity map $i: (X, \tau_1) \to (X, \tau_2)$ is continuous if:
A: $\tau_1 \subset \tau_2$
B: $\tau_2 \subset \tau_1$
C: $\tau_1 \cap \tau_2 = \emptyset$
D: Always
Correct Answer: B
Q26: In a discrete space, the closure of any set $A$ is:
A: $X$
B: $A$
C: $\emptyset$
D: $\text{Bd}(A)$
Correct Answer: B
Q27: A set is closed if and only if it contains all its ______.
A: Interior points
B: Limit points
C: Isolated points
D: Exterior points
Correct Answer: B
Q28: The union of two closed sets is always:
A: Open
B: Closed
C: Dense
D: Empty
Correct Answer: B
Q29: The intersection of an arbitrary collection of open sets is:
A: Always open
B: Not necessarily open
C: Always closed
D: Always empty
Correct Answer: B
Q30: A constant function $f(x) = k$ is always:
A: Continuous
B: Open
C: A Homeomorphism
D: Bijective
Correct Answer: A
Q31: If $f: X \to Y$ is continuous and $A \subset X$, then $f(\overline{A}) \subset \overline{f(A)}$. This is:
A: Always True
B: Always False
C: True only for open maps
D: True only for isometries
Correct Answer: A
Q32: A map $f: X \to Y$ is a homeomorphism if it is open, continuous, and ______.
A: Injective only
B: Surjective only
C: Bijective
D: Constant
Correct Answer: C
Q33: In the standard topology of $\mathbb{R}$, which set is dense?
A: $\mathbb{Z}$ (Integers)
B: $\mathbb{Q}^c$ (Irrationals)
C: $\{0, 1\}$
D: Finite sets
Correct Answer: B
Q34: The number of topologies on a singleton set $\{a\}$ is:
A: 0
B: 1
C: 2
D: Infinite
Correct Answer: B
Q35: Which space is homeomorphic to the interval $(0, 1)$?
A: $\mathbb{R}$
B: $[0, 1]$
C: $S^1$ (Circle)
D: $\mathbb{Q}$
Correct Answer: A
Q36: A sub-base $\mathcal{S}$ for $\mathbb{R}$ with usual topology is the collection of all:
A: Open intervals $(a, b)$
B: Semi-infinite intervals $(-\infty, a)$ and $(b, \infty)$
C: Closed intervals $[a, b]$
D: Singletons
Correct Answer: B
Q37: If $X$ is discrete and $Y$ is any space, every map $f: X \to Y$ is ______.
A: Continuous
B: Closed
C: Open
D: Bijective
Correct Answer: A
Q38: If $Y$ is indiscrete and $X$ is any space, every map $f: X \to Y$ is ______.
A: Bijective
B: Continuous
C: Constant
D: Closed
Correct Answer: B
Q39: The composition of two homeomorphisms is a:
A: Constant map
B: Homeomorphism
C: Closed map only
D: Linear map
Correct Answer: B
Q40: A space with the cofinite topology is second countable if and only if $X$ is:
A: Infinite
B: Countable
C: Uncountable
D: Discrete
Correct Answer: B
Q41: Interior and Closure are related by: $\text{Int}(A) = $
A: $X \setminus \overline{(X \setminus A)}$
B: $\overline{X \setminus A}$
C: $X \cup \overline{A}$
D: $\text{Bd}(A)$
Correct Answer: A
Q42: The property of being First Countable depends on the existence of a countable base at ______.
A: The origin
B: Each point
C: Infinity
D: Open sets
Correct Answer: B
Q43: A subspace $Y \subset X$ inherits the ______ topology.
A: Discrete
B: Relative (or induced)
C: Indiscrete
D: Product
Correct Answer: B
Q44: A set $A$ is nowhere dense if $\text{Int}(\overline{A}) = $:
A: $X$
B: $\emptyset$
C: $A$
D: $\text{Bd}(A)$
Correct Answer: B
Q45: The Sorgenfrey Line refers to $\mathbb{R}$ with the ______ topology.
A: Standard
B: Lower Limit
C: Upper Limit
D: Cofinite
Correct Answer: B
Q46: Every metric space satisfies the ______ countability axiom.
A: First
B: Second
C: Zero
D: Third
Correct Answer: A
Q47: If a space is Second Countable, it is also ______.
A: Compact
B: First Countable
C: Discrete
D: Finite
Correct Answer: B
Q48: The intersection of an open set and a closed set is:
A: Open
B: Closed
C: Can be neither
D: Always empty
Correct Answer: C
Q49: Isometry implies homeomorphism, but homeomorphism ______ implies isometry.
A: Always
B: Never
C: Not necessarily
D: Usually
Correct Answer: C
Q50: Which map is NOT always continuous?
A: Identity map
B: Constant map
C: Projection map
D: Inclusion map
Correct Answer: A
Q51: A point $x$ is an interior point of $A$ if $A$ is a ______ of $x$.
A: Subset
B: Neighborhood
C: Boundary
D: Complement
Correct Answer: B
Q52: The empty set $\emptyset$ is:
A: Open only
B: Closed only
C: Both open and closed
D: Neither
Correct Answer: C
Q53: If $\text{Bd}(A) = \emptyset$, then $A$ is:
A: Empty
B: Clopen
C: Dense
D: Discrete
Correct Answer: B
Q54: The topology generated by the basis of all open balls is the ______ topology.
A: Metric
B: Cofinite
C: Indiscrete
D: Product
Correct Answer: A
Q55: A function $f$ is continuous at $x_0$ if for every nbd $V$ of $f(x_0)$, there exists a nbd $U$ of $x_0$ such that:
A: $f(U) \subset V$
B: $f(V) \subset U$
C: $f^{-1}(U) \subset V$
D: $f(U) = V$
Correct Answer: A
Q56: In a $T_1$ space, every singleton set is ______.
A: Open
B: Closed
C: Dense
D: Compact
Correct Answer: B
Q57: The set of isolated points of $A$ is $A \setminus$ ______.
A: $\text{Int}(A)$
B: $A'$ (derived set)
C: $\overline{A}$
D: $\emptyset$
Correct Answer: B
Q58: If $A \subset B$, then $\text{Int}(A)$ ______ $\text{Int}(B)$.
A: $\subset$
B: $\supset$
C: $=$
D: $\cap$
Correct Answer: A
Q59: A space is Lindelöf if every open cover has a ______ subcover.
A: Finite
B: Countable
C: Empty
D: Dense
Correct Answer: B
Q60: The derived set of $\mathbb{Z}$ in $\mathbb{R}$ with usual topology is:
A: $\mathbb{Z}$
B: $\emptyset$
C: $\mathbb{R}$
D: $\{0\}$
Correct Answer: B
Q61: Uniform continuity is preserved under ______.
A: Homeomorphism
B: Isometry
C: Addition only
D: Any continuous map
Correct Answer: B
Q62: The closure of $(0, 1)$ in $\mathbb{R}$ with discrete topology is:
A: $[0, 1]$
B: $(0, 1)$
C: $\{0, 1\}$
D: $\emptyset$
Correct Answer: B
Q63: Every finite set in a $T_1$ space is ______.
A: Open
B: Closed
C: Dense
D: Homeomorphic
Correct Answer: B
Q64: A basis $\mathcal{B}$ for $\tau$ must cover $X$, meaning $\cup \mathcal{B} = $:
A: $\emptyset$
B: $X$
C: $\tau$
D: $\mathcal{P}(X)$
Correct Answer: B
Q65: The topology having the fewest open sets is ______.
A: Discrete
B: Indiscrete
C: Standard
D: Cofinite
Correct Answer: B
Q66: In the usual topology on $\mathbb{R}$, $\text{Int}(\mathbb{Q}) = $:
A: $\mathbb{Q}$
B: $\emptyset$
C: $\mathbb{R}$
D: $\mathbb{Q}^c$
Correct Answer: B
Q67: A map $f: X \to Y$ is closed if images of closed sets are ______.
A: Open
B: Closed
C: Empty
D: Homeomorphic
Correct Answer: B
Q68: A collection $\mathcal{B}$ is a basis if for any $B_1, B_2 \in \mathcal{B}$, their intersection is a ______ of elements of $\mathcal{B}$.
A: Union
B: Intersection
C: Complement
D: Difference
Correct Answer: A
Q69: If $X$ is finite, the cofinite topology on $X$ is ______.
A: Standard
B: Discrete
C: Indiscrete
D: Non-existent
Correct Answer: B
Q70: The set of points $x$ such that every nbd of $x$ intersects $A$ is the ______ of $A$.
A: Interior
B: Closure
C: Boundary
D: Exterior
Correct Answer: B
Q71: The function $f(x) = e^x$ is a homeomorphism from $\mathbb{R}$ to ______.
A: $\mathbb{R}$
B: $(0, \infty)$
C: $[0, \infty)$
D: $\mathbb{Z}$
Correct Answer: B
Q72: Topological spaces $(X, \tau_1)$ and $(Y, \tau_2)$ are homeomorphic if there exists a ______ between them preserving openness.
A: Bijection
B: Injection
C: Surjection
D: Constant
Correct Answer: A
Q73: The derived set of $(0, 1)$ in standard $\mathbb{R}$ is:
A: $(0, 1)$
B: $[0, 1]$
C: $\{0, 1\}$
D: $\emptyset$
Correct Answer: B
Q74: Every second countable space is ______.
A: Separable
B: Discrete
C: Finite
D: Indiscrete
Correct Answer: A
Q75: A space is separable if it has a countable ______ subset.
A: Open
B: Dense
C: Closed
D: Finite
Correct Answer: B
Q76: In the cofinite topology, any two non-empty open sets have a ______ intersection.
A: Empty
B: Non-empty
C: Finite
D: Singleton
Correct Answer: B
Q77: A space $X$ is $T_0$ if at least one point has a neighborhood ______ the other.
A: Containing
B: Excluding
C: Equal to
D: Disjoint from
Correct Answer: B
Q78: The boundary of $\mathbb{Q}$ in $\mathbb{R}$ is:
A: $\emptyset$
B: $\mathbb{R}$
C: $\mathbb{Q}$
D: $\mathbb{Q}^c$
Correct Answer: B
Q79: Isometry maps open balls to ______.
A: Closed sets
B: Open balls
C: Rectangles
D: Singletons
Correct Answer: B
Q80: The set of limit points of a set $A$ is closed? (In standard $\mathbb{R}$)
A: Always
B: Never
C: Sometimes
D: Only for finite sets
Correct Answer: A
Q81: Uniformly continuous maps preserve ______ sequences in metric spaces.
A: Convergent
B: Cauchy
C: Divergent
D: Constant
Correct Answer: B
Q82: A sub-base is always ______ than a base.
A: Larger
B: Smaller (or equal)
C: Equal
D: Finer
Correct Answer: B
Q83: The closure of the empty set is always ______.
A: $X$
B: $\emptyset$
C: A singleton
D: Undefined
Correct Answer: B
Q84: The interior of $X$ is always ______.
A: $\emptyset$
B: $X$
C: Boundary
D: Discrete
Correct Answer: B
Q85: The union of two dense sets is ______ dense.
A: Always
B: Never
C: Not necessarily
D: Only if open
Correct Answer: A
Q86: If $\tau$ is the discrete topology, then $(X, \tau)$ is ______ countable if $X$ is countable.
A: First
B: Second
C: Both A and B
D: Neither
Correct Answer: C
Q87: The standard topology on $\mathbb{R}$ is ______ than the cofinite topology.
A: Coarser
B: Finer
C: Equal
D: Incomparable
Correct Answer: B
Q88: A bijective map that is continuous but not open is ______ a homeomorphism.
A: Always
B: Never
C: Sometimes
D: Defined as
Correct Answer: B
Q89: The intersection of two dense sets is ______ dense.
A: Always
B: Not necessarily
C: Never
D: Only if they are open
Correct Answer: B
Q90: Metric spaces are always ______ spaces.
A: $T_2$ (Hausdorff)
B: Discrete
C: Indiscrete
D: Finite
Correct Answer: A
Q91: Which of these is NOT a base for $\mathbb{R}$?
A: All open intervals
B: All open intervals with rational endpoints
C: All closed intervals
D: All intervals $(a, \infty)$ and $(-\infty, b)$
Correct Answer: C
Q92: The boundary of a closed set $A$ is ______ $A$.
A: Disjoint from
B: Contained in
C: Equal to
D: Larger than
Correct Answer: B
Q93: The boundary of an open set $A$ is ______ $A$.
A: Contained in
B: Disjoint from
C: Equal to
D: Dense in
Correct Answer: B
Q94: A map $f$ is continuous if $f(Cl(A)) \subset Cl(f(A))$. This is a ______ condition.
A: Necessary only
B: Sufficient only
C: Necessary and sufficient
D: False
Correct Answer: C
Q95: A homeomorphism preserves the ______ of the space.
A: Geometry
B: Topological structure
C: Size
D: Calculus
Correct Answer: B
Q96: The set of points $x$ where $f(x)=g(x)$ for continuous $f,g$ into a $T_2$ space is ______.
A: Open
B: Closed
C: Dense
D: Empty
Correct Answer: B
Q97: The diameter of a set is a ______ property.
A: Topological
B: Metric
C: Algebraic
D: Discrete
Correct Answer: B
Q98: The derived set of a finite set in a $T_1$ space is ______.
A: The set itself
B: $\emptyset$
C: Infinite
D: X
Correct Answer: B
Q99: Every second countable space is ______.
A: First countable
B: Discrete
C: Metric
D: Indiscrete
Correct Answer: A
Q100: In a topological space, the interior of the interior of $A$ is equal to:
A: $\overline{A}$
B: $\text{Int}(A)$
C: $A$
D: $\text{Bd}(A)$
Correct Answer: B