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		{\bf \Large Abstract }\\
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		{\Large   An Introduction to {\TeX} and  Explanation of the Structure  in Vector Spaces
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		{\large By:\\
			$\mathbb{R}$oosta\TeX }
		
		%{\Large\bf Title of the Thesis ...........................................................................................}\\
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		Latex is a powerful typesetting system widely used for academic and scientific documents. 
		It is especially well suited for writing mathematical formulas, 
		technical papers and  strutured texts.
		We will introduce a kind of 
		probabilistic dynamic epistemic logic which is adapted 
		to solve the Monty Hall dilemma and will prove its completeness. 
		Then we will obtain a formal solution for this dilemma within this logic.
		
		Let X and Y be topological spaces. If C is a compact subspace of X and U is an open subset of Y, define 
		$ S(C,U)=\{f\lvert f\in C(X,Y)\ and\ f(C)\subseteq U\}. $
		The sets $ S(C,U)$ form a subbasis for a topology on $C(X,Y) $ that is called the compact-open topology. In particular case if $ Y=\mathbb{R} $ we denoted by $ C_{c}(X)$.
		Actually in this thesis we show, among other results, if X is a locally compact [locally compact, separable and satisfying the first countability axiom] space, then the following results are equivalent:\\
		a) $C_{c}(X)$ has countable [bounding] tightness. b) $C_{c}(X)$ is a Frechet-Urysohn space.\\ c) $C_{c}(X)$ is a Frechet space. 
		d) $C_{c}(X)$ contains a dense (LM) subspace. e) X is $\sigma-{\rm compact}$.
		

		
		
		
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		IN THE NAME OF GOD 
		
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		{\LARGE  
		An Introduction to {\TeX} and 
		 Explanation of the Structure  in Vector Spaces
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		{\Large By:
			
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			THESIS  \\
			SUBMITTED TO THE DEPARTMENT 
			OF MATHEMATICS \\
			OF  \\
			SHIRAZ UNIVERSITY OF TECHNOLOGY  \\
			IN PARTIAL  \\
			FULFILLMENT 
			OF THE REQUIREMENTS FOR \\
			THE DEGREE OF MASTER OF SCIENCE (M.Sc.) \\
			IN  \\
			APPLIED MATHEMATICS   
			
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			 {\bf  Thesis committee members: ~Excellent }  \\
			Mohammadi  M., Assistant Prof. of Math. (Supervisor)   \\
		Mohammadian M., Prof. of Math. (Advisor)      \\
			Mohammadi M., Associate Prof. of Math. (Referee)   \\
		Mohammadi M., Associate Prof. of Math. (Referee)   \\
			
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	\hfil 	Sep 2025 
	
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	{ \large
		
	Shiraz University of Technology \\
	Department of Mathematics  
			
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	M.Sc. Thesis 
	
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		An Introduction to {\TeX} and  
		Explanation of the Structure  in Vector Spaces
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	By:  
	
	{\Large  	$\mathbb{R}$oosta\TeX  }
	
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	Supervisor:  

	 {\Large Donald Knuth }
	
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	Advisor:  
	
 	{\Large  ... }
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	September 2025

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