M.Sc. Mathematics - Topology Exam Prep

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Topology - Unit I: Topology, Space, Continuity

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Q1: A collection $\tau$ of subsets of a set $X$ is a topology on $X$ if it satisfies which condition?
Q2: In the discrete topology on $X$, every subset is:
Q3: The indiscrete topology on $X$ contains exactly:
Q4: If $\tau_1 \subset \tau_2$, we say $\tau_1$ is ______ than $\tau_2$.
Q5: In the cofinite topology on an infinite set $X$, a set $A$ is open if $X \setminus A$ is:
Q6: The intersection of any collection of topologies on $X$ is:
Q7: The closure of a set $A$, denoted $\overline{A}$, is the intersection of all ______ sets containing $A$.
Q8: A point $x$ is a limit point of $A$ if every neighborhood of $x$ contains at least one point of $A$ ______ $x$.
Q9: The interior of $A$ is the ______ open set contained in $A$.
Q10: A set $A$ is dense in $X$ if:
Q11: The set of rational numbers $\mathbb{Q}$ is ______ in $\mathbb{R}$ with the usual topology.
Q12: A collection $\mathcal{B}$ is a base for $\tau$ if every open set is a ______ of elements of $\mathcal{B}$.
Q13: A collection $\mathcal{S}$ is a sub-base if finite ______ of its elements form a base.
Q14: A space $X$ is called Second Countable if it has a ______ base.
Q15: The boundary of $A$ is defined as $\text{Bd}(A) = $:
Q16: A map $f: X \to Y$ is continuous if for every open set $V \subset Y$, $f^{-1}(V)$ is ______ in $X$.
Q17: A function $f$ is a homeomorphism if it is a continuous bijection and its ______ is also continuous.
Q18: If $f$ is a homeomorphism, then $X$ and $Y$ are called ______ spaces.
Q19: A map $f$ is called an 'open map' if the image of every open set is ______.
Q20: An isometry between metric spaces preserves:
Q21: Every isometry is a ______ map.
Q22: Uniform continuity is a concept that requires a ______ structure.
Q23: Which of the following is a topological property (invariant under homeomorphism)?
Q24: In $\mathbb{R}$ with the usual topology, the interior of the set $[0, 1]$ is:
Q25: The identity map $i: (X, \tau_1) \to (X, \tau_2)$ is continuous if:
Q26: In a discrete space, the closure of any set $A$ is:
Q27: A set is closed if and only if it contains all its ______.
Q28: The union of two closed sets is always:
Q29: The intersection of an arbitrary collection of open sets is:
Q30: A constant function $f(x) = k$ is always:
Q31: If $f: X \to Y$ is continuous and $A \subset X$, then $f(\overline{A}) \subset \overline{f(A)}$. This is:
Q32: A map $f: X \to Y$ is a homeomorphism if it is open, continuous, and ______.
Q33: In the standard topology of $\mathbb{R}$, which set is dense?
Q34: The number of topologies on a singleton set $\{a\}$ is:
Q35: Which space is homeomorphic to the interval $(0, 1)$?
Q36: A sub-base $\mathcal{S}$ for $\mathbb{R}$ with usual topology is the collection of all:
Q37: If $X$ is discrete and $Y$ is any space, every map $f: X \to Y$ is ______.
Q38: If $Y$ is indiscrete and $X$ is any space, every map $f: X \to Y$ is ______.
Q39: The composition of two homeomorphisms is a:
Q40: A space with the cofinite topology is second countable if and only if $X$ is:
Q41: Interior and Closure are related by: $\text{Int}(A) = $
Q42: The property of being First Countable depends on the existence of a countable base at ______.
Q43: A subspace $Y \subset X$ inherits the ______ topology.
Q44: A set $A$ is nowhere dense if $\text{Int}(\overline{A}) = $:
Q45: The Sorgenfrey Line refers to $\mathbb{R}$ with the ______ topology.
Q46: Every metric space satisfies the ______ countability axiom.
Q47: If a space is Second Countable, it is also ______.
Q48: The intersection of an open set and a closed set is:
Q49: Isometry implies homeomorphism, but homeomorphism ______ implies isometry.
Q50: Which map is NOT always continuous?
Q51: A point $x$ is an interior point of $A$ if $A$ is a ______ of $x$.
Q52: The empty set $\emptyset$ is:
Q53: If $\text{Bd}(A) = \emptyset$, then $A$ is:
Q54: The topology generated by the basis of all open balls is the ______ topology.
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:
Q56: In a $T_1$ space, every singleton set is ______.
Q57: The set of isolated points of $A$ is $A \setminus$ ______.
Q58: If $A \subset B$, then $\text{Int}(A)$ ______ $\text{Int}(B)$.
Q59: A space is Lindelöf if every open cover has a ______ subcover.
Q60: The derived set of $\mathbb{Z}$ in $\mathbb{R}$ with usual topology is:
Q61: Uniform continuity is preserved under ______.
Q62: The closure of $(0, 1)$ in $\mathbb{R}$ with discrete topology is:
Q63: Every finite set in a $T_1$ space is ______.
Q64: A basis $\mathcal{B}$ for $\tau$ must cover $X$, meaning $\cup \mathcal{B} = $:
Q65: The topology having the fewest open sets is ______.
Q66: In the usual topology on $\mathbb{R}$, $\text{Int}(\mathbb{Q}) = $:
Q67: A map $f: X \to Y$ is closed if images of closed sets are ______.
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 $\mathcal{B}$.
Q69: If $X$ is finite, the cofinite topology on $X$ is ______.
Q70: The set of points $x$ such that every nbd of $x$ intersects $A$ is the ______ of $A$.
Q71: The function $f(x) = e^x$ is a homeomorphism from $\mathbb{R}$ to ______.
Q72: Topological spaces $(X, \tau_1)$ and $(Y, \tau_2)$ are homeomorphic if there exists a ______ between them preserving openness.
Q73: The derived set of $(0, 1)$ in standard $\mathbb{R}$ is:
Q74: Every second countable space is ______.
Q75: A space is separable if it has a countable ______ subset.
Q76: In the cofinite topology, any two non-empty open sets have a ______ intersection.
Q77: A space $X$ is $T_0$ if at least one point has a neighborhood ______ the other.
Q78: The boundary of $\mathbb{Q}$ in $\mathbb{R}$ is:
Q79: Isometry maps open balls to ______.
Q80: The set of limit points of a set $A$ is closed? (In standard $\mathbb{R}$)
Q81: Uniformly continuous maps preserve ______ sequences in metric spaces.
Q82: A sub-base is always ______ than a base.
Q83: The closure of the empty set is always ______.
Q84: The interior of $X$ is always ______.
Q85: The union of two dense sets is ______ dense.
Q86: If $\tau$ is the discrete topology, then $(X, \tau)$ is ______ countable if $X$ is countable.
Q87: The standard topology on $\mathbb{R}$ is ______ than the cofinite topology.
Q88: A bijective map that is continuous but not open is ______ a homeomorphism.
Q89: The intersection of two dense sets is ______ dense.
Q90: Metric spaces are always ______ spaces.
Q91: Which of these is NOT a base for $\mathbb{R}$?
Q92: The boundary of a closed set $A$ is ______ $A$.
Q93: The boundary of an open set $A$ is ______ $A$.
Q94: A map $f$ is continuous if $f(Cl(A)) \subset Cl(f(A))$. This is a ______ condition.
Q95: A homeomorphism preserves the ______ of the space.
Q96: The set of points $x$ where $f(x)=g(x)$ for continuous $f,g$ into a $T_2$ space is ______.
Q97: The diameter of a set is a ______ property.
Q98: The derived set of a finite set in a $T_1$ space is ______.
Q99: Every second countable space is ______.
Q100: In a topological space, the interior of the interior of $A$ is equal to:
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