# Tagged Questions

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### Analogue of closed graph theorem

This is the analogue of closed graph theorem for compact space Suppose that $X$ and $K$ are metric spaces, that $K$ is compact, and that the graph of $f: X \rightarrow K$ is a closed subset ...
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### Detail on a theorem of continuity and compactness

I have come across a proof in a book. I have trouble convincing myself on a statement on said proof. The theorem is a well-known one. I am stating the version I found on Ross's "Elementary Analysis." ...
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### About the continuity of a function in the closed graph theorem proof

I'm reading Functional Analysis book of Rudin, and in the proof of the closed graph theorem, there's one point that I don't understand. Can someone please explain it to me? I really appreciate this. ...
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### if a map and its inverse are continuous, does that imply injection?

I've proved that a mapping of one topology to another and its inverse are both continuous. so since f and f inverse are continuous, can I therefore say that they're injective?
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### Prove B is a closed subset of X given the f and g are continous?

Let $(X;\rho)$, $(Y;\sigma)$ be metric spaces. Let $f,g : X \to Y$ be continuous. Prove that the set $B=\{x\in X: f(x)=g(x)\}$ is a closed subset of $X$
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### Prove that the identity map $(C[0,1],d_1) \rightarrow (C[0,1],d_\infty)$ is not continuous

$$d_\infty = \max|x_i - y_i|$$ $$d_1 = \sum_{i=1}^n |x_i - y_i|$$ The first part of this question was to prove that the identity map $$(C[0,1],d_\infty) \rightarrow (C[0,1],d_1)$$ is continuous, ...
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### $\sup_n \inf_{y \in X} (F(y) + n d(x,y))= F(x)$

Let $(X,d)$ be a metric space and $F : X \rightarrow [0, +\infty)$ a lower semicontinuous function. Then $$\sup_n \inf_{y \in X} (F(y) + n d(x,y))= F(x).$$ Is this true? Intuitively it works since ...
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### Proving that half an isometry is a homeomorphism

Let $(K,d)$ be a compact metric space and $f:K\rightarrow K$ such that $$\forall x \in K, \forall y \in K, d(f(x),f(y)) \geq d(x,y)$$ Prove that $f$ is a homeomorphism. What I managed to prove is ...
### Isn't the result true for any $A\subset X?$
There's a problem in my text which reads as: Let $f: (X, d)\to(Y, d)$ be continuous. Let $A\subset X$ be open. Show that the restriction $f|_A$ of $f$ to $A$ is a continuous function from the metric ...