# Tagged Questions

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### Suppose $x$ is a limit point of $A \subset X$, then if $f: A \to Y$ is continuous, is it true that $f(x)$ is a limit point of $f(A)$?

So I already know that a counterexample is $f(x) = c$ for $c$ is a constant, but I can't seem to prove this statement by contradiction, all I did was go back and forth. "Proof": If $f(x)$ ...
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### Use contraction mapping theorem to prove

Help! I am taking a math course, and I just can't figure out this proof: Let $\alpha,\beta\in R^n$, $a\in R$, and $A$ be an $n\times n$ nonsingular matrix. Use contraction mapping theorem to prove ...
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### Prove that a defined function g is continuous for a certain point

Consider the set of rational numbers $\mathbb{Q}$ equipped with the subspace topology inherited from $\mathbb{R}$. Let $c \in \mathbb{R}$. Define the function $g_{c}: \mathbb{Q} \to \mathbb{Q}$ via: ...
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### Prove that a function of the rational numbers $\mathbb{Q}$ with subspace topology inherited from $\mathbb{R}$ is injective

Consider the set of rational numbers $\mathbb{Q}$ equipped with the subspace topology inherited from $\mathbb{R}$. Suppose $g: \mathbb{R} \to \mathbb{R}$ and $h: \mathbb{R} \to \mathbb{R}$ are ...
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### Does continuity of $f$ imply $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$?

I'm struggling to prove or disprove that the continuity of $f$ implies $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$. $f:X\to Y$ is a map between metric spaces $(X,d),(Y,d')$ while $\bar M$ denotes the ...
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### Proving properties of closures using intersection of indexed sets and topology

How would I write a proof for this example? Let X be a set, and let $B \subseteq \mathcal P \left({X}\right)$. Define $B^* =${ $U \subseteq X:$ There is an index set $I$ and $U_{i} \in B$ for each ...
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### Question about proofs with topological spaces

How would I write a proof for this example? Let $(X, \tau_{1})$, $(Y, \tau_{2})$ and $(Z, \tau_{3})$ be topological spaces. A function ${f}: X \rightarrow Y$ is said to be continuous if for every V ...
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### finite subset topology

Let $X$ be a set and $\tau=\left\{X\right\}\,\cup\,\{u_i\subset X|u_i \text{ is finite}\}$. Is $\tau$ is a topology on $X$? My effort to show this is as follows: 1) $X\in\tau$ by definition and ...
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### Metric Space, Induced topology

I'm trying to show that the metric topology is indeed a topology. To do so, I want to show the following three statements are true: $\emptyset$ and X are in $\tau$ finite intersections of open sets ...
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### Could someone help me to improve the proof writing?

I will prove the following claim. I'm not a native English speaker. Could someone help me to improve the writing? A regular pseudocompact Moore space is ccc and first countable. Prove: I will ...
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### Prove that if a set $E$ is closed iff it's complement $E^{c}$ is open

I was wondering if this proof was right. $\Leftarrow$ Suppose $E$ is closed. Then choose $x\in E^{c}$, then $x\notin E$, and so $x$ is not a limit point of $E$. Hence there exists a neighborhood ...
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### Trying to prove noncompactness over the discrete metric

I wrote this version of proof. But would like some feedback. I am proving that the discrete metric $(X,d)$ is not compact. $X$ is infinite, it contains infinitely many points Let $x_i \in X$, then ...
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### Why is the induction proof not sufficient? Topology…

I went to a exercise class and I got really confused. Consider the problem: Let $\lbrace A_{n} \rbrace$ be a sequence of connected subspaces of $X$, such that $A_{n}\cap A_{n+1}\neq \emptyset$ for ...
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### Why is this last step even necessary in this proof with open sets?

Let $E^o$ denote the set of all interior point of a set $E$. Prove that $E^o$ is always open. Proof: For $p \in E^o$, there is a neighborhood $N_r (p) \subset E$. Since neighborhoods are open, for ...
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