hcju/proofwikips huggingface dataset:
Existence of Maximal Compact Topological Space which is not Hausdorff
Let $S$ be a set. Let $\tau$ be the minimal subset of the power set $\powerset S$ such that $\struct {S, \tau}$ is a compact topological space. Then it is not necessarily the case that $\struct {S, \tau}$ is a Hausdorff space.
Gauss's Lemma (Polynomial Theory)
Gauss's lemma on polynomials may refer to any of the following statements.
Infinite Product of Sigma-Compact Spaces is not always Sigma-Compact
Let $I$ be an indexing set with infinite cardinality. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let each of $\struct {S_\alpha, \tau_\alpha}$ be $\sigma$-compact. Then it is not necessarily the case that $\struct {S, \tau}$ is also $\sigma$-compact.
Uncountable Product of Sequentially Compact Spaces is not always Sequentially Compact
Let $I$ be an indexing set with uncountable cardinality. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let each of $\struct {S_\alpha, \tau_\alpha}$ be sequentially compact. Then it is not necessarily the case that $\struct {S, \tau}$ is also sequentially compact.
Product of Countably Compact Spaces is not always Countably Compact
Let $I$ be an indexing set. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let each of $\struct {S_\alpha, \tau_\alpha}$ be countably compact. Then it is not necessarily the case that $\struct {S, \tau}$ is also countably compact.
Product of Lindelöf Spaces is not always Lindelöf
Let $I$ be an indexing set. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let each of $\struct {S_\alpha, \tau_\alpha}$ be a Lindelöf space. Then it is not necessarily the case that $\struct {S, \tau}$ is also Lindelöf space.
Uncountable Product of Second-Countable Spaces is not always Second-Countable
Let $I$ be an indexing set with uncountable cardinality. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let each of $\struct {S_\alpha, \tau_\alpha}$ be a second-countable space. Then it is not necessarily the case that $\struct {S, \tau}$ is also a second-countable space.
Paracompactness is not always Preserved under Open Continuous Mapping
Let $T_A = \struct {X_A, \tau_A}$ be a topological space which is paracompact. Let $T_B = \struct {X_B, \tau_B}$ be another topological space. Let $\phi: T_A \to T_B$ be a mapping which is both continuous and open. Then it is not necessarily the case that $T_B$ is also paracompact.
Correspondence Theorem for Ring Epimorphisms
Let $A$ and $B$ be commutative rings with unity. Let $\pi : A \to B$ be a ring epimorphism. Let $I$ be the set of ideals of $A$ containing the kernel $\operatorname{ker} \pi$. Let $J$ be the set of ideals of $B$.
Union of Closure with Closure of Complement is Whole Space
Let $T = \left({S, \tau}\right)$ be a topological space. Let $H \subseteq S$ be a subset of $S$. Let $H^-$ denote the closure of $H$ in $T$. Let $S \setminus H$ denote the complement of $H$ relative to $S$. Then: :$H^- \cup \left({S \setminus H}\right)^- = S$
Equivalent Characterisations of Irrational Periodic Continued Fraction
Let $x \in \R \setminus \Q$ be an irrational number. Let $(a_n)_{n\geq 0}$ be its continued fraction. Let $N \geq 0$ be a natural number.  :$(1):\quad$ The sequence of partial quotients $(a_n)_{n\geq 0}$ is periodic for $n \geq N$. :$(2):\quad$ The sequence of complete quotients $(x_n)_{n\geq 0}$ is periodic for $n \geq N$. :$(3):\quad$ There exists $M > N$ such that $x_M = x_N$.
Component of Point is not always Intersection of its Clopen Sets
Let $T = \struct {S, \tau}$ be a topological space. Let $x \in S$. Let $\map {\operatorname {Comp}_x} T$ denote the component of $x$ in $T$. Let $K_x = \displaystyle \bigcap_{x \mathop \in K} K$ clopen in $T$. Then it is not always the case that $\map {\operatorname {Comp}_x} T = K_x$
Complement of Clopen Set is Clopen
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a clopen set of $T$. Let $\relcomp S H$ denote the complement of $H$ relative to $S$. Then $\relcomp S H$ is also a clopen set of $T$.
Clopen Set and Complement form Separation
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a clopen set of $T$. Let $\relcomp S H$ be the complement of $H$ relative to $S$. Then $H$ and $\relcomp S H$ form a separation of $T$.
Path Component is not necessarily Arc Component
Let $T = \struct {S, \tau}$ be a topological space. Let $P$ be a path component of $T$. Then it is not necessarily the case that $P$ is also an arc component of $T$.
Component is not necessarily Path Component
Let $T = \struct {S, \tau}$ be a topological space. Let $C$ be a component of $T$. Then it is not necessarily the case that $C$ is also an path component of $T$.
Quasicomponent is not necessarily Component
Let $T = \struct {S, \tau}$ be a topological space. Let $Q$ be a quasicomponent of $T$. Then it is not necessarily the case that $C$ is also a component of $T$.
Simple Infinite Continued Fraction is Uniquely Determined by Limit
Let $(a_n)_{n\geq 0}$ and $(b_n)_{n\geq 0}$ be simple infinite continued fractions in $\R$. Let $(a_n)_{n\geq 0}$ and $(b_n)_{n\geq 0}$ have the same limit. Then they are equal.
Continued Fraction Expansion of Irrational Number Converges to Number Itself
Let $x$ be an irrational number. Then the continued fraction expansion of $x$ converges to $x$.
Continued Fraction Expansion of Limit of Simple Infinite Continued Fraction equals Expansion Itself
Let Let $(a_n)_{n\geq 0}$ be a simple infinite continued fractions in $\R$. Then $(a_n)_{n\geq 0}$ converges to an irrational number, whose continued fraction expansion is $(a_n)_{n\geq 0}$.
Correspondence between Irrational Numbers and Simple Infinite Continued Fractions
Let $\R \setminus \Q$ be the set of irrational numbers. Let $S$ be the set of all simple infinite continued fractions in $\R$. The mappings: :$\R \setminus \Q \to S$ that sends an irrational number to its continued fraction expansion :$S \to \R \setminus \Q$ that sends a simple infinite continued fractions to its value are inverses of each other.
Ultraconnected Space is Connected
Let $T = \struct {S, \tau}$ be a topological space which is ultraconnected. Then $T$ is connected.
Irreducible Space is not necessarily Path-Connected
Let $T = \left({S, \tau}\right)$ be a topological space which is irreducible. Then $T$ is not necessarily path-connected.
Ultraconnected Space is not necessarily Arc-Connected
Let $T = \struct {S, \tau}$ be a topological space which is ultraconnected. Then $T$ is not necessarily arc-connected.
Correspondence between Rational Numbers and Simple Finite Continued Fractions
Let $\Q$ be the set of rational numbers. Let $S$ be the set of all simple finite continued fractions in $\Q$, whose last partial quotient is not $1$. The mappings: :$\Q \to S$ that sends an rational number to its continued fraction expansion :$S \to \Q$ that sends a simple finite continued fractions to its value are inverses of each other.
Existence of Non-Locally Connected Space where Components and Quasicomponents are Equal
There exists at least one example of a topological space which is not locally connected, but whose components and quasicomponents are equal.
Locally Connected Space is not necessarily Connected
Let $T = \struct {S, \tau}$ be a topological space which is locally connected. Then it is not necessarily the case that $T$ is also a connected space.
Irreducible Space with Finitely Many Open Sets is Path-Connected
Let $T = \left({S, \tau}\right)$ be a irreducible topological space. Let its topology $\tau$ be finite. Then $T$ is path-connected.
Topological Space with Generic Point is Path-Connected
Let $T = \struct {S, \tau}$ be a topological space. Let $T$ have a generic point $g \in S$. Then $T$ is path-connected.
Floor of Simple Finite Continued Fraction
Let $\sequence {a_k}_{k \mathop \ge 0}$ be a simple finite continued fraction of length $n \ge 0$. Let $x = [a_0, \ldots, a_n]$ be its value. Then the floor of $x$ is the partial denominator $a_0$: :$\floor x = a_0$ unless $n = 1$ and $a_1 = 1$, in which case $x = \floor x = a_0 + 1$.
Value of Finite Continued Fraction of Strictly Positive Real Numbers is Strictly Positive
Let $(a_0, \ldots, a_n)$ be a finite continued fraction in $\R$ of length $n \geq 0$. Let all partial quotients $a_k>0$ be strictly positive. Let $x = [a_0, a_1, \ldots, a_n]$ be its value. Then $x>0$.
Value of Finite Continued Fraction of Real Numbers is at Least First Term
Let $(a_0, \ldots, a_n)$ be a finite continued fraction in $\R$ of length $n \geq 0$. Let the partial quotients $a_k>0$ be strictly positive for $k>0$. Let $x = [a_0, a_1, \ldots, a_n]$ be its value. Then $x \geq a_0$, and $x>a_0$ if the length $n\geq 1$.
Finite Simple Continued Fraction has Rational Value
Let $n \ge 0$ be a natural number. Let $\paren {a_0, \ldots, a_n}$ be a simple finite continued fraction of length $n$. Then its value $\sqbrk {a_0, \ldots, a_n}$ is a rational number.
Accuracy of Convergents of Convergent Simple Infinite Continued Fraction
Let $C = (a_0, a_1, \ldots)$ be an infinite simple continued fraction in $\R$. Let $C$ converge to $x \in \R$. For $n\geq0$, let $C_n = p_n/q_n$ be the $n$th convergent of $C$, where $p_n$ and $q_n$ are the $n$th numerator and denominator. Then for all $n\geq 0$: :$\left\vert x - \dfrac {p_n}{q_n}\right\vert < \dfrac 1{q_nq_{n+1}}$.
Locally Path-Connected Space is not necessarily Locally Arc-Connected
Let $T = \struct {S, \tau}$ be a topological space which is locally path-connected. Then it is not necessarily the case that $T$ is also a locally arc-connected space.
Path-Connected Space is not necessarily Locally Path-Connected
Let $T = \struct {S, \tau}$ be a topological space which is path-connected. Then it is not necessarily the case that $T$ is also locally path-connected.
Lower Bounds for Denominators of Simple Continued Fraction
Let $n \in \N \cup \{\infty\}$ be an extended natural number. Let $\left[{a_0, a_1, a_2, \ldots}\right]$ be a simple continued fraction in $\R$ of length $N$. Let $q_0, q_1, q_2, \ldots$ be its denominators.
Equality of Rational Numbers
Let $a, b, c, d$ be integers, with $b$ and $d$ nonzero.  :$(1): \quad$ The rational numbers $\dfrac a b$ and $\dfrac c d$ are equal. :$(2): \quad$ The integers $a d$ and $b c$ are equal.
Locally Path-Connected Space is not necessarily Path-Connected
Let $T = \struct {S, \tau}$ be a topological space which is locally path-connected. Then it is not necessarily the case that $T$ is also path-connected.
Arc-Connected Space is not necessarily Locally Arc-Connected
Let $T = \struct {S, \tau}$ be a topological space which is arc-connected. Then it is not necessarily the case that $T$ is also locally arc-connected.
Locally Arc-Connected Space is not necessarily Arc-Connected
Let $T = \struct {S, \tau}$ be a topological space which is locally arc-connected. Then it is not necessarily the case that $T$ is also arc-connected.
Odd Convergents of Simple Continued Fraction are Strictly Decreasing
The odd convergents satisfy $C_1 > C_3 > C_5 > \cdots$
Even Convergents of Simple Continued Fraction are Strictly Increasing
The even convergents satisfy $C_0 < C_2 < C_4 \cdots$.
Denominators of Simple Continued Fraction are Strictly Positive
Let $n \in \N \cup \{\infty\}$ be an extended natural number. Let $(a_0, a_1, \ldots)$ be a simple continued fraction in $\R$ of length $n$. Let $q_0, q_1, q_2, \ldots$ be its denominators. Then for $0 \leq k \leq n$ we have $q_k > 0$.
Simple Finite Continued Fraction is Almost Determined by Value
Let $n,m \geq 0$ be natural number. Let $(a_k)_{0 \leq k \leq m}$ and $(b_k)_{0 \leq k \leq n}$ be simple finite continued fractions in $\R$. Let $(a_k)_{0 \leq k \leq m}$ and $(b_k)_{0 \leq k \leq n}$ have the same value. Then either: *$n = m$, and the sequences are equal. *$n = m + 1$, $a_k = b_k$ for $k \leq m$, $a_m = b_m-1$ and $b_{m+1} = 1$ *$m = n + 1$, $a_k = b_k$ for $k \leq n$, $b_n = a_n-1$ and $a_{n+1} = 1$
Subgroup Generated by One Element is Set of Powers
Let $G$ be a group. Let $a \in G$. Then the subgroup generated by $a$ is the set of powers: :$\gen a  = \set {a^n : n \in \Z}$
Finite Product Space is Connected iff Factors are Connected
Let $T_1 = \struct {S_1, \tau_1}, T_2 = \struct {S_2, \tau_2}, \dotsc, T_n = \struct {S_n, \tau_n}$ be topological spaces. Let $T = \displaystyle \prod_{i \mathop = 1}^n T_i$ be the product space of $T_1, T_2, \ldots, T_n$. Then $T$ is connected  each of $T_1, T_2, \ldots, T_n$ are connected.
Binary Coproduct in Preadditive Category is Biproduct
Let $A$ be a preadditive category. Let $a_1, a_2$ be objects of $A$. Let $(a_1 \sqcup a_2, i_1, i_2)$ be their binary coproduct, assuming it exists. Let $p_1 : a_1 \sqcup a_2 \to a_1$ be the unique morphism with: :$p_1 \circ i_1 = 1 : a_1 \to a_1$ :$p_1 \circ i_2 = 0 : a_1 \to a_2$ Let $p_2 : a_1 \sqcup a_2 \to a_2$ be the unique morphism with: :$p_2 \circ i_1 = 0 : a_2 \to a_1$ :$p_2 \circ i_2 = 1 : a_2 \to a_2$ where $1$ is the identity morphism and $0$ is the zero morphism. Then $(a_1 \sqcup a_2, i_1, i_2, p_1, p_2)$ is the binary biproduct of $a_1$ and $a_2$.
Continuous Mapping from Compact Space to Hausdorff Space Preserves Local Connectedness
Let $T_1 = \struct {S_1, \tau_1}$ be a compact topological space. Let $T_2 = \struct {S_2, \tau_2}$ be a $T_2$ (Hausdorff) space. Let $f: T_1 \to T_2$ be a continuous mapping. Let $T_1$ be locally connected. Then $T_2$ is also locally connected.
Minimum Degree Bound for Simple Planar Graph
Let $G$ be a simple connected planar graph. Then:  :$\map \delta G \le 5$ where $\map \delta G$ denotes the minimum degree of vertices of $G$.
Linear Bound Lemma
For a simple connected planar graph $G_n$, where $n \geq 3$ is a number of vertices: :$m \leq 3 n − 6$, where $m$ is a number of edges.
Existence of Connected Space which is Totally Pathwise Disconnected
There exists at least one example of a topological space which is both connected and totally pathwise disconnected.
Zero Dimensional Space is not necessarily T0
Let $T = \struct {S, \tau}$ be a zero dimensional topological space. Then $T$ is not necessarily a $T_0$ (Kolmogorov) space.
Scattered Space is not necessarily T1
Let $T = \struct {S, \tau}$ be a scattered topological space. Then $T$ is not necessarily a $T_1$ (Fréchet) space.
Existence of Connected Non-T1 Scattered Space
There exists at least one example of a connected topological space which is not a $T_1$ (Fréchet) space, which is also a scattered space.
Sorgenfrey Line is not Second-Countable
Let $T = \struct {\mathbb R, \tau}$ be the Sorgenfrey line. Then $T$ is not second-countable.
Existence of Biconnected Set without Dispersion Point
There exists at least one example of a biconnected set which does not have a dispersion point.
Existence of Connected Punctiform Space
There exists at least one example of a connected topological space which is also punctiform.
Metric Space is Completely Normal
Let $M = \struct {A, d}$ be a metric space. Then $M$ is a completely normal space.
Bounded Metric Space is not necessarily Totally Bounded
Let $M = \struct {A, d}$ be a bounded metric space. Then it is not necessarily the case that $M$ is totally bounded.
Total Boundedness is not Preserved under Homeomorphism
Let $M = \struct {A, d}$ be a totally bounded metric space. Let $M' = \struct {A', d'}$ be a metric space. Let $M$ be homeomorphic to $M'$. Then it is not necessarily the case that $M'$ is totally bounded.
Topological Completeness is not Hereditary
Let $T = \struct {S, \tau}$ be a topological space which is topologically complete. Let $H \subseteq S$ be a subset of $S$. Let $\struct {H, \tau_H}$ be the topological subspace of $T$ induced by $H$. Then it is not necessarily the case that $\struct {H, \tau_H}$ is also topologically complete. That is, topological completeness is not hereditary.
Completion Theorem (Metric Space)/Lemma 4
:$\tilde M = \struct {\tilde A, \tilde d}$ is unique up to isometry.
Sorgenfrey Line is First-Countable
Let $\R$ be the set of real numbers. Let $\BB = \set {\hointr a b: a, b \in \R}$. Let $\tau$ be the topology generated by $\BB$, that is, the Sorgenfrey line. Then $\tau$ is first-countable.
Metrizable Space is not necessarily Second-Countable
Let $T = \struct {S, \tau}$ be a topological space which is metrizable. Then it is not necessarily the case that $T$ is second-countable.
Regular Paracompact Space is not necessarily Metrizable
Let $T = \struct {S, \tau}$ be a topological space which is regular and paracompact. Then it is not necessarily the case that $T$ is metrizable.
Uniform Space whose Topology is Metrizable is not necessarily Metrizable
Let $\UU$ be a uniformity on a set $S$. Let $\struct {\struct {S, \UU}, \tau}$ be the uniform space generated from $\UU$. Let $T = \struct {S, \tau}$ be the uniformizable space yielded by $\struct {\struct {S, \UU}, \tau}$. Let $T$ be a metrizable space. Then it is not necessarily the case that $\UU$ is itself a metrizable uniformity.
Open Sets in Indiscrete Topology
$H$ is an open set of $T$  either $H = S$ or $H = \O$.
Closed Sets in Indiscrete Topology
$H$ is a closed set of $T$  either $H = S$ or $H = \O$.
F-Sigma Sets in Indiscrete Topology
$H$ is an $F_\sigma$ ($F$-sigma) set of $T$  either $H = S$ or $H = \O$.
Subset of Indiscrete Space is Compact
$H$ is compact in $T$.
Partition Topology is not Completely Hausdorff
Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons. Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$. Then $T$ is not a $T_{2 \frac 1 2}$ (completely Hausdorff) space.
Partition Topology is not Hausdorff
Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons. Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$. Then $T$ is not a $T_2$ (Hausdorff) space.
Partition Topology is not T1
Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons. Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$. Then $T$ is not a $T_1$ (Fréchet) space.
Odd-Even Topology is Lindelöf
$T$ is Lindelöf.
Odd-Even Topology is Separable
$T$ is separable.
Odd-Even Topology is First-Countable
$T$ is first-countable.
Particular Point Space less Particular Point is Discrete
Let $T = \left({S, \tau_p}\right)$ be a particular point space, whose particular point is $p$. Let $H = S \setminus \left\{ {p}\right\}$ where $\setminus$ denotes set difference. Then the topological subspace $T_H = \left({H, \tau_H}\right)$ induced on $H$ by $\tau_p$ is a discrete space.
Zero is Accumulation Point of Sequence in Sierpiński Space
Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space. The sequence in $T$: :$\sigma = \sequence {0, 1, 0, 1, \ldots}$ has $0$ as an accumulation point.
Infinite Particular Point Space is not Countably Paracompact
Let $T = \left({S, \tau_p}\right)$ be an infinite particular point space. Then $T$ is not countably paracompact.
Closed Extension Topology is not Hausdorff
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$. Then $T^*_p$ is not a $T_2$ (Hausdorf) space.
Closed Extension Topology is not T3
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$. Then $T^*_p$ is not a $T_3$ space.
Either-Or Topology is T4
Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is a $T_4$ space.
Limit Points in Fort Space
Let $T = \struct {S, \tau_p}$ be a Fort space. Let $x \in S$ such that $x \ne p$. Then $p$ is the only limit point of $x$.
Rational Numbers in Real Euclidean Plus Space are Open Set
Let $\R$ be the set of real numbers. Let $d: \R \times \R \to \R$ be the Euclidean plus metric: :$\map d {x, y} := \size {x - y} + \displaystyle \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\max_{j \mathop \le i} \frac 1 {\size {x - r_j} } - \max_{j \mathop \le i} \frac 1 {\size {y - r_j} } } }$ Let $\Q$ be the set of rational numbers. Then $\Q$ is an open set of $\struct {\R, d}$.
Alexandroff Extension is Compact
Let $T = \struct {S, \tau}$ be a non-empty topological space. Let $p$ be a new element not in $S$. Let $S^* := S \cup \set p$. Let $T^* = \struct {S^*, \tau^*}$ be the Alexandroff extension on $S$. Then $T^*$ is a compact topological space.
Condition for Alexandroff Extension to be T1 Space
Let $T = \struct {S, \tau}$ be a non-empty topological space. Let $p$ be a new element not in $S$. Let $S^* := S \cup \set p$. Let $T^* = \struct {S^*, \tau^*}$ be the Alexandroff extension on $S$. Then $T^*$ is a $T_1$ (Fréchet) space  $T$ is a $T_1$ (Fréchet) space.
Condition for Alexandroff Extension to be T2 Space
Let $T = \struct {S, \tau}$ be a non-empty topological space. Let $p$ be a new element not in $S$. Let $S^* := S \cup \set p$. Let $T^* =\struct {S^*, \tau^*}$ be the Alexandroff extension on $S$. Then $T^*$ is a $T_2$ (Hausdorff) space  $T$ is a locally compact Hausdorff space.
Alexandroff Extension which is T2 Space is also T4 Space
Let $T = \struct {S, \tau}$ be a non-empty topological space. Let $p$ be a new element not in $S$. Let $S^* := S \cup \set p$. Let $T^* = \struct {S^*, \tau^*}$ be the Alexandroff extension on $S$. Let $T^*$ be a $T_2$ (Hausdorff) space. Then $T^*$ is a $T_4$ space.
Alexandroff Extension of Rational Number Space is not Hausdorff
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Let $p$ be a new element not in $\Q$. Let $\Q^* := \Q \cup \set p$. Let $T^* = \struct {\Q^*, \tau^*}$ be the Alexandroff extension on $\struct {\Q, \tau_d}$. Then $T^*$ is not a $T_2$ (Hausdorff) space.
Alexandroff Extension of Rational Number Space is T1 Space
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Let $p$ be a new element not in $\Q$. Let $\Q^* := \Q \cup \set p$. Let $T^* = \struct {\Q^*, \tau^*}$ be the Alexandroff extension on $\struct {\Q, \tau_d}$. Then $T^*$ is a $T_1$ (Fréchet) space.
Particular Point of Alexandroff Extension of Rational Number Space is Dispersion Point
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Let $p$ be a new element not in $\Q$. Let $\Q^* := \Q \cup \set p$. Let $T^* = \struct {\Q^*, \tau^*}$ be the Alexandroff extension on $\left({\Q, \tau_d}\right)$. Then $p$ is a dispersion point of $T^*$.
Alexandroff Extension of Rational Number Space is Connected
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Let $p$ be a new element not in $\Q$. Let $\Q^* := \Q \cup \set p$. Let $T^* = \struct {\Q^*, \tau^*}$ be the Alexandroff extension on $\struct {\Q, \tau_d}$. Then $T^*$ is a connected space.