hcju/proofwikips huggingface dataset:
Definition:Limit Superior/Definition 1
Let $L$ be the set of all real numbers which are the limit of some subsequence of $\sequence {x_n}$. From Existence of Maximum and Minimum of Bounded Sequence, $L$ has a maximum. This maximum is called the '''limit superior'''. It can be denoted: :$\displaystyle \map {\limsup_{n \mathop \to \infty} } {x_n} = \overline l$
Definition:Limit Superior/Definition 2
The '''limit superior of $\sequence {x_n}$''' is defined and denoted as: :$\displaystyle \map {\limsup_{n \mathop \to \infty} } {x_n} = \inf \set {\sup_{m \mathop \ge n} x_m: n \in \N}$
Definition:Limit Inferior/Definition 1
Let $L$ be the set of all real numbers which are the limit of some subsequence of $\sequence {x_n}$. From Existence of Maximum and Minimum of Bounded Sequence, $L$ has a minimum. This minimum is called the '''limit inferior'''. It can be denoted: :$\displaystyle \map {\liminf_{n \mathop \to \infty} } {x_n} = \underline l$
Definition:Limit Inferior/Definition 2
The '''limit inferior of $\sequence {x_n}$''' is defined and denoted as: :$\displaystyle \map {\liminf_{n \mathop \to \infty} } {x_n} = \sup \set {\inf_{m \mathop \ge n} x_m: n \in \N}$
Definition:Chi-Squared Distribution
Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $\Img X = \hointr 0 \infty$.  Let $r$ be a strictly positive integer. $X$ is said to have a '''chi-squared distribution''' with $r$ degrees of freedom  it has probability density function: :$\displaystyle \map {f_X} x = \dfrac 1 {2^{r / 2} \map \Gamma {r / 2} } x^{\paren {r / 2} - 1} e^{- x / 2}$ where $\Gamma$ denotes the gamma function. This is written:  :$X \sim \chi^2_r$
Definition:Operations Research
'''Operations research''' is the branch of applied mathematics which studies the management of human organizations.
Definition:Chi Distribution
Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $\Img X = \hointr 0 \infty$.  Let $r$ be a strictly positive integer. $X$ is said to have a '''chi distribution''' with $r$ degrees of freedom  it has probability density function: :$\displaystyle \map {f_X} x = \dfrac 1 {2^{\paren {r / 2} - 1} \map \Gamma {r / 2} } x^{r - 1} e^{- x^2 / 2}$ where $\Gamma$ denotes the gamma function. This is written:  :$X \sim \chi_r$
Definition:F-Distribution
Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $\Img X = \hointr 0 \infty$.  Let $n, m$ be strictly positive integers. $X$ is said to have an '''F-distribution''' with $\tuple {n, m}$ degrees of freedom  it has probability density function: :$\displaystyle \map {f_X} x = \frac {m^{m / 2} n^{n / 2} x^{\paren {n / 2} - 1} } {\paren {m + n x}^{\paren {n + m} / 2} \map \Beta {n / 2, m / 2} }$ where $\Beta$ denotes the beta function.  This is written:  :$X \sim F_{n, m}$
Definition:Constant/Linguistic Note
The word '''constant''' can be used either as a {{WP|Noun|noun}}: :''Let $c$ be a constant between $0$ and $1$'' or as an {{WP|Adjective|adjective}}: :''Let $c$ be a constant real number between $0$ and $1$'' Which is intended can usually be deduced from the context.
Definition:Chess/Chessboard
A '''chessboard''' is an array (usually square) of alternating dark and light squares (conventionally referred to as '''black''' and '''white''', even though they may well not be those actual colours). {{ChessDiagram| | |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   | }} This may be referred to as an $m \times n$ '''chessboard''', where: :$m$ is the number of ranks :$n$ is the number of files. For the conventional '''chessboard''', as depicted above, $m = n = 8$. === Rank === {{:Definition:Chess/Chessboard/Rank}} === File === {{:Definition:Chess/Chessboard/File}} === Square === {{:Definition:Chess/Chessboard/Square}}
Definition:Hypothesis (Inductive Statistics)
A '''hypothesis''' is a statement about a population parameter.
Definition:Chess/Chessboard/Rank
A '''row''' of a '''chessboard''' is known as a '''rank'''. It is identified by '''number''': :$1$ is the '''rank''' nearest white :The '''ranks''' are numbered sequentially to the '''rank''' nearest black.
Definition:Chess/Player
The two players in a game of chess are identified by the colour of the pieces they are using. The sets of pieces are made from materials of two contrasting colours, conventionally referred to as '''black''' and '''white'''. Most everyday chess sets actually ''are'' '''black''' and '''white'''. === White === {{:Definition:Chess/Player/White}} === Black === {{:Definition:Chess/Player/Black}} A diagram of a chessboard is conventionally positioned on a page such that '''white''' is at the bottom, playing upwards towards '''black''', and '''black''' is at the top, playing downwards towards '''white'''.
Definition:Chess/Player/White
The '''chess player''' who is using the lighter-coloured pieces is conventionally referred to as '''white'''.
Definition:Chess/Player/Black
The '''chess player''' who is using the darker-coloured pieces is conventionally referred to as '''black'''.
Definition:Chess/Chessboard/File
A '''column''' of a '''chessboard''' is known as a '''file'''. It is identified by '''letter''' :$\text a$ is the '''file''' on the edge of the board to the left of white (that is, to the right of black) :The '''files''' are lettered sequentially up to the '''file''' on the edge of the board to the left of black (that is, to the right of white).
Definition:Chess/Chessboard/Square
The individual spaces of the chessboard are referred to as its '''squares'''. They are conventionally identified by a letter-number pair, where: :the letter identifies its file :the number identifies its rank.
Definition:Chess/Piece/King
The '''king''' File:Chess klt45.svg can be moved one square in any direction, including diagonally. It cannot be moved into a position where it is in check.
Definition:Chess/Rules/Castling
'''Castling''' is a special move where, in one move, the king moves two squares towards a rook, and the rook jumps over the king and lands on the next square the other side of the king. This can only be done if: :$(1): \quad$ The king and rook are both in their initial positions. :$(2): \quad$ The squares in between the king and rook are empty. :$(3): \quad$ The king is not in check, and neither of the $2$ squares it moves through are under attack.
Definition:Chess/Piece/Queen
The '''queen''' File:Chess qlt45.svg can be moved any number of squares in any of $8$ directions: forwards, backwards, left, right, and the $4$ diagonals. It cannot be moved past a square occupied by another piece.
Definition:Chess/Draw/Insufficient Pieces
If neither player has sufficient pieces to force checkmate, then the game of chess is '''drawn'''.
Definition:Chess/Rules/Under Attack
If a chess piece $A$ may legally move to a square which is occupied by a piece $B$ of the opposite colour, $B$ is said to be '''under attack''' by $A$. In the same way, any square to which $A$ may legally move may also be said to be '''under attack'''.
Definition:Chess/Rules/In Check
Uniquely among the chess pieces the king may not legally move into a square which is under attack by a piece of the opposite colour. That is, a king may not be moved so as to be placed under attack. If player $a$ moves a piece so as to place the king of player $b$ under attack, then player $b$ is said to be '''in check'''. Player $a$ is then obliged to say '''"Check."''' Player $b$ must make a move so that his or her king is no longer under attack. Such a move by player $b$ is known as '''getting out of check'''.
Definition:Chess/Rules/Checkmate
If player $a$ has put player $b$'s king in check, but player $b$ has no legal move that will get him or her out of check, then player $a$ has won. This situation is known as '''checkmate'''. Player $a$ is said to have '''checkmated''' player $b$.
Definition:Chess/Rules
With the exception of the knight, no chess piece may move past a square which is occupied by another piece. === Under Attack === {{:Definition:Chess/Rules/Under Attack}} === Capture === {{:Definition:Chess/Rules/Capture}} === Pawn === {{:Definition:Chess/Rules/Capture/Pawn}} === Castling === {{:Definition:Chess/Rules/Castling}} === In Check === {{:Definition:Chess/Rules/In Check}} === Checkmate === {{:Definition:Chess/Rules/Checkmate}}
Definition:Chess/Piece/Bishop
The '''bishop''' File:Chess blt45.svg can be moved any number of squares in any of the $4$ diagonal directions. It cannot be moved past a square occupied by another piece.
Definition:Chess/Piece/Rook
The '''rook''' File:Chess rlt45.svg can be moved any number of squares in any of the $4$ orthogonal directions: up, down, left or right. It cannot be moved past a square occupied by another piece.
Definition:Chess/Piece/Knight
The '''knight''' File:Chess nlt45.svg can be moved two squares in any orthogonal direction (up, down, left or right), and then one square at right angles to that direction. It passes over intervening pieces and does not stop until reaching the end of its move. It cannot be moved onto a square occupied by another piece of the same colour.
Definition:Chess/Pawn/Promotion
If a pawn moves onto the square on the final rank, it is then exchanged for any other type of piece which replaces the pawn on that square. This is called '''promotion'''. The player of that pawn decides which piece to replace it with, which may be neither another pawn nor a king. If it is promoted to a queen (which is usual, the queen being the most powerful piece), this operation is colloquially referred to as '''queening'''.
Definition:Chess/Pawn
A '''pawn''' File:Chess plt45.svg may move in the following modes. $(1): \quad$ It may move one square towards the opposing player,  that square is empty. $(2): \quad$ If it has not yet been moved, and  both of the squares is empty, it may move $2$ squares forward instead of $1$. Some authorities do not classify the pawn as a piece. === Promotion === {{:Definition:Chess/Pawn/Promotion}}
Definition:Chess/Rules/Capture/Pawn/Normal
If one of the $2$ diagonally adjacent squares towards the opposing player is occupied by an opposing piece, the pawn may move into that square and capture that piece.
Definition:Chess/Rules/Capture/Pawn/En Passant
Let pawn $a$ be on the player's $5$th rank. Let pawn $b$ be of the opposite colour to pawn $a$. Let pawn $b$ be on its starting position, on one of the files adjacent to the one occupied by pawn $a$. Let pawn $b$ move forward $2$ spaces, in the process crossing over one of the squares which is under attack from pawn $a$. Then pawn $a$, '''on its next move only''', may move into that square crossed over by pawn $b$, and capture pawn $b$ "while it is passing". This mode of capture is known as '''capture ''en passant'''''.
Definition:Chess/Rules/Capture/Pawn
A capture in chess by a pawn works completely differently from that by any other piece. It may happen in one of $2$ circumstances. ==== Normal Pawn Capture ==== {{:Definition:Chess/Rules/Capture/Pawn/Normal}} ==== En Passant Pawn Capture ==== {{:Definition:Chess/Rules/Capture/Pawn/En Passant}}
Definition:Chess/Piece
There are a number of different kinds of '''chess piece'''. They are distinguished by their modes of movement, as follows: === King === {{:Definition:Chess/Piece/King}} === Queen === {{:Definition:Chess/Piece/Queen}} === Bishop === {{:Definition:Chess/Piece/Bishop}} === Knight === {{:Definition:Chess/Piece/Knight}} === Rook === {{:Definition:Chess/Piece/Rook}} === Pawn === {{:Definition:Chess/Pawn}}
Definition:Chess/Starting Position
The '''starting position''' of a game of chess is: {{ChessDiagram| | |rd|nd|bd|qd|kd|bd|nd|rd |pd|pd|pd|pd|pd|pd|pd|pd |  |  |  |  |  |  |  |    |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |pl|pl|pl|pl|pl|pl|pl|pl |rl|nl|bl|ql|kl|bl|nl|rl | }} Note the following: :$(1): \quad$ The top left square is white :$(2): \quad$ The queens start on squares of their own colours :$(3): \quad$ White is always first to move.
Definition:Chess/Move
Each turn in a game of chess consists of transferring a piece from one square to another, according to the rules. Such a transfer is known as a '''move'''. There is one exception: the operation of '''castling''' involves the moving of $2$ pieces.
Definition:Retrograde Analysis
'''Retrograde analysis''' is a technique used to determine the sequence of chess moves which were played that led to a given position.
Definition:Hypothesis Test
A '''hypothesis test''' is a rule that specifies, for a null hypothesis $H_0$ and alternative hypothesis $H_1$:  * For which sample values the decision is made to accept $H_0$. * For which sample values $H_0$ is rejected and $H_1$ is accepted.
Definition:Erlang Distribution
Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $\Img X = \hointr 0 \infty$.  Let $k$ be a strictly positive integer. Let $\lambda$ be a strictly positive real number. $X$ is said to have an '''Erlang distribution''' with parameters $k$ and $\lambda$  it has probability density function: :$\map {f_X} x = \dfrac {\lambda^k x^{k - 1} e^{- \lambda x} } {\map \Gamma k}$ where $\Gamma$ denotes the gamma function. This is written:  :$X \sim \map {\operatorname {Erlang} } {k, \lambda}$
Definition:Laplace Distribution
Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $\Img X = \R$.  Let $\mu$ be a real number.  Let $b$ be a strictly positive real number. $X$ is said to have a '''Laplace distribution''' with parameters $\mu$ and $b$  it has probability density function: :$\map {f_X} x = \dfrac 1 {2 b} \map \exp {- \dfrac {\size {x - \mu} } b}$ This is written:  :$X \sim \map {\operatorname {Laplace} } {\mu, b}$
Definition:Bijection/Technical Note
{{LatexFor|for = f: S \leftrightarrow T}} {{LatexFor|for = f: S \cong T}} {{LatexFor|for = S \stackrel f \cong T}}
Definition:Simple Hypothesis
Let $\theta$ be a population parameter of some population.  Let the parameter space of $\theta$ be $\Omega$.  Let $\Omega_0$ and $\Omega_1$ be disjoint subsets of $\Omega$ such that $\Omega_0 \cup \Omega_1 = \Omega$.  Consider the hypotheses:  :$H_0: \theta \in \Omega_0$ :$H_1: \theta \in \Omega_1$ We call $H_i$, for $i \in \set {0, 1}$, a '''simple hypothesis''' if $\Omega_i$ contains only a single element.
Definition:Composite Hypothesis
Let $\theta$ be a population parameter of some population.  Let the parameter space of $\theta$ be $\Omega$.  Let $\Omega_0$ and $\Omega_1$ be disjoint subsets of $\Omega$ such that $\Omega_0 \cup \Omega_1 = \Omega$.  Consider the hypotheses:  :$H_0: \theta \in \Omega_0$ :$H_1: \theta \in \Omega_1$ We call $H_i$, for $i \in \set {0, 1}$, a '''composite hypothesis''' if $\Omega_i$ contains more than a single element.
Definition:Critical Region
Let $\theta$ be a population parameter of some population $P$.  Let $\Omega$ be the parameter space of $\theta$.  Let $\mathbf X$ be a random sample from $P$.  Let $T = \map f {\mathbf X}$ be a sample statistic. Let $\delta$ be a test procedure of the form:  :reject $H_0$ if $T \in C$ for some null hypothesis $H_0$ and some $C \subset \Omega$. We refer to $C$ as the '''critical region''' of $\delta$.
Definition:Test Statistic
Let $\theta$ be a population parameter of some population $P$.  Let $\Omega$ be the parameter space of $\theta$.  Let $\mathbf X$ be a random sample from $P$.  Let $T = \map f {\mathbf X}$ be a sample statistic. Let $\delta$ be a test procedure of the form:  :reject $H_0$ if $T \in C$ for some null hypothesis $H_0$ and some $C \subset \Omega$. We refer to $T$ as the '''test statistic''' of $\delta$.
Definition:Power Function
Let $\theta$ be a population parameter of some population. Let $\Omega$ be the parameter space of $\theta$.    Let $\delta$ be a test procedure for hypotheses about the value of $\theta$.  Let $C$ be the critical region of $\delta$.  Let $T$ be the test statistic of $\delta$.  The '''power function''' of $\delta$, written $\map \pi {\theta_0}$, is defined by: :$\map \pi {\theta_0} = \map \Pr {T \in C \mid \theta = \theta_0 }$ for all $\theta_0 \in \Omega$.
Definition:Parameter Space
Let $\theta$ be a population parameter of some population. The '''parameter space''' of $\theta$, typically denoted $\Omega$, is the domain of $\theta$. That is, it is the set of all possible values that $\theta$ can take.
Definition:Type I Error
A '''type I error''' is the rejection of a true null hypothesis as the result of a test procedure.
Definition:Type II Error
A '''type II error''' is the failure to reject a false null hypothesis as the result of a test procedure.
Definition:Size (Inductive Statistics)
Let $\theta$ be a population parameter of some population. Let $\Omega$ be the parameter space of $\theta$.  Let $\Omega_0$ and $\Omega_1$ be disjoint subsets of $\Omega$ such that $\Omega_0 \cup \Omega_1 = \Omega$. Let $\delta$ be a test procedure of the hypotheses: :$H_0: \theta \in \Omega_0$ :$H_1: \theta \in \Omega_1$ Let $\pi$ be the power function of $\delta$.  The '''size''' of $\delta$, usually denoted $\alpha$, is defined as:  :$\displaystyle \alpha = \sup_{\theta \in \Omega_0} \map \pi \theta$
Definition:Covariance
Let $X$ and $Y$ be random variables.  Let $\mu_X = \expect X$ and $\mu_Y = \expect Y$, the expectations of $X$ and $Y$ respectively, exist and be finite. Then the '''covariance''' of $X$ and $Y$ is defined by:  :$\cov {X, Y} = \expect {\paren {X - \mu_X} \paren {Y - \mu_Y} }$ where this expectation exists.
Definition:Pearson Correlation Coefficient
Let $X$ and $Y$ be random variables.  Let the variances of $X$ and $Y$ exist and be finite. Then the '''Pearson correlation coefficient''' of $X$ and $Y$, typically denoted $\map \rho {X, Y}$, is defined by:  :$\map \rho {X, Y} = \dfrac {\cov {X, Y} } {\sqrt {\var X \, \var Y} }$ where $\cov {X, Y}$ is the covariance of $X$ and $Y$.
Definition:Transversality Conditions
Let $\map {\mathbf y} x$ be a differentiable vector-valued function. Let $J \sqbrk {\mathbf y}$ be a functional of the following form: :$\ds J \sqbrk {\mathbf y} = \int_{P_1}^{P_2} \map F {x, \mathbf y, \mathbf y', \ldots} \rd x$ where $P_1$, $P_2$ are points on given differentiable manifolds $M_1$ and $M_2$. Suppose we are looking for $\mathbf y$ extremizing $J$. The system of equations to be solved consists of differential Euler equations and algebraic equations at both endpoints. Then the set of all algebraic equations at both endpoints are called '''transversality conditions'''.
Definition:Unbiased Estimator
Let $\theta$ be a population parameter of some statistical model. Let $\delta$ be an estimator of $\theta$. We call $\delta$ an '''unbiased estimator''' if its bias is equal to $0$ regardless of the true value of $\theta$.
Definition:Hardy-Littlewood Maximal Function
{{MissingLinks|throughout}} {{explain|presumably the "Hardy-Littlewood Maximal Function" is that function which is returned by the Hardy-Littlewood maximal operator? More precision needed here.}} The Hardy-Littlewood maximal operator takes a locally integrable function $f: \R^d \to \R$  and returns another function $M f$ that, at each point $x \in \R^d$ gives the maximum average value that $f$ can have on balls centered at that point.  More precisely, :$\displaystyle \map {M f} x := \sup_{r \mathop > 0} \frac 1 {\size {\map B {x, r} } } \int_{\map B {x, r} } \size {\map f y} \rd y$ where: :$\map B {x, r}$  is the ball of radius $r$ centered at $x$ :$\size E$ denotes the Lebesgue measure of $E \subset \R^d$.
Definition:Precisely One Function
Let $p_1, p_2, \ldots, p_n$ be statements. The '''precisely one function''' is the propositional function $\map P {p_1, p_2, \ldots, p_n}$ defined as: :$\map P {p_1, p_2, \ldots, p_n}$ is true  '''precisely one''' of $p_1, p_2, \ldots, p_n$ is true.
Definition:Differential Equation/Total
A '''total differential equation''' is a '''differential equation''' which contains: :more than one dependent variable :one independent variable which may or may not appear explicitly in that differential equation.
Definition:Differential Equation/Degree
Let $f$ be a '''differential equation''' which can be expressed as a polynomial in all the derivatives involved. The '''degree''' of $f$ is defined as being the power to which the derivative of the highest order is raised. By default, if not specifically mentioned, the '''degree''' of a '''differential equation''' is assumed to be $1$.
Definition:Alexandrian School
The '''Alexandrian School''' was active in the city of {{WP|Alexandria|Alexandria}} for the approximate millennium leading up to the city's destruction in $641$ C.E. It stood as the cultural hub of the Greek sphere of influence. The {{WP|Musaeum|Alexandrian Musaeum}} stood at the centre of this hub, and has been argued as being the archetypal forerunner of the modern University. It had a library which, at its peak, is believed to have contained over $700 \, 000$ volumes. After the fall of {{WP|Alexandria|Alexandria}}, most of its scholars migrated to {{WP|Constantinople|Constantinople}}, preserving the ancient Greek mathematical learning, where it remained for some $800$ more years until the {{WP|Renaissance|Renaissance}}.
Definition:Convolution of Probability Distributions
Let $X$ and $Y$ be independent random variables.  Let $Z = X + Y$.  The probability distribution of $Z$ is called the '''convolution''' of the probability distributions of $X$ and $Y$.
Definition:Convergence in Probability
Let $\sequence {X_n}_{n \ge 1}$ be a sequence of random variables. Let $X$ be a random variable.  We say that $\sequence {X_n}$ '''converges in probability''' to $X$ if:  :$\displaystyle \lim_{n \to \infty} \map \Pr {\size {X_n - X} < \varepsilon} = 1$ for all real $\varepsilon > 0$. This is written:  :$X_n \xrightarrow p X$
Definition:Convergence in Distribution
Let $\sequence {X_n}_{n \ge 1}$ be a sequence of random variables. Let $X$ be a random variable.  We say that $\sequence {X_n}$ '''converges in distribution''' to $X$ if:  :$\displaystyle \lim_{n \mathop \to \infty} \map \Pr {X_n \le x} = \map \Pr {X \le x}$ for all $x$ for which the map $x \mapsto \map \Pr {X \le x}$ is continuous. This is written:  :$X_n \xrightarrow d X$
Definition:Random Sample
Let $X_i$ be a random variable with $\Img {X_i} = \Omega$, for all $1 \le i \le n$. Let $F_i$ be the cumulative distribution function of $X_i$ for all $1 \le i \le n$.  We say that $X_1, X_2, \ldots, X_n$ form a '''random sample''' of size $n$ if:  :$X_i$ and $X_j$ are independent if $i \ne j$ :$\map {F_1} x = \map {F_i} x$ for all $x \in \Omega$ for all $1 \le i, j \le n$. If $X_1, X_2, \ldots, X_n$ form a '''random sample''', they are said to be '''independent and identically distributed''', commonly abbreviated '''i.i.d'''.
Definition:Almost Sure Convergence
Let $\sequence {X_n}_{n \ge 1}$ be a sequence of random variables. Let $X$ be a random variable.  We say that $\sequence {X_n}$ '''almost surely converges''' to $X$ if:  :$\displaystyle \map \Pr {\lim_{n \mathop \to \infty} \size {X_n - X} < \varepsilon} = 1$ for all real $\varepsilon > 0$. This is written: :$X_n \xrightarrow {\text {a.s.}} X$
Definition:Path Component/Equivalence Class
Let $\sim$ be the equivalence relation on $T$ defined as: :$x \sim y \iff x$ and $y$ are path-connected. The equivalence classes of $\sim$ are called the '''path components of $T$'''. If $x \in T$, then the '''path component of $T$''' containing $x$ (that is, the set of points $y \in T$ with $x \sim y$) can be denoted by $\map {\operatorname{PC}_x} T$.
Definition:Path Component/Union of Path-Connected Sets
The '''path component of $T$ containing $x$''' is defined as: : $\displaystyle \map {\operatorname{PC}_x} T = \bigcup \left\{{A \subseteq S: x \in A \land A}\right.$ is path-connected $\left.\right\}$
Definition:Path Component/Maximal Path-Connected Set
The '''path component of $T$ containing $x$''' is defined as: :the maximal path-connected set of $T$ that contains $x$.