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### Continuity and adherence in topology [duplicate]

Do we have $f:(E,\tau)\to(F,\sigma)$ continuous if and only if $\forall A\subset E, f(\overline{A})= \overline{f(A)}$ or just: $f:(E,\tau)\to(F,\sigma)$ continuous if and only if \forall A\...
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### Topology: Homeomorphism between finite complement topology in $\mathbb{R}$ and one of its subspaces

My class notes say that because $U=\mathbb{R}\backslash\{x_1,x_2,..,x_n\}$ has the same cardinality than $\mathbb{R}$, there exists a homeomorphism between: $(U,T_{cof})$ and $(\mathbb{R},T_{cof})$, ...
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### Continuity and interior

I have questions about the relation between continuity and interior based on the article ;A map is continuous if and only if for every set, the image of closure is contained in the closure of image ...
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### Prove that $f(x) = 0$ for all $x ∈ R$. [duplicate]

Let $f(x)$ be a continuous function such that $f(r) = 0$ for all rational numbers r. Prove that $f(x) = 0$ for all $x ∈ R$.
### How can a continuous function induce a proper inclusion $f(\overline{A})\subsetneq \overline{f(A)}$?
Let $f:(X, d_X)\longrightarrow (Y, d_Y)$ be a continuous function between two metric spaces, $A\subseteq X$. We have $f(\overline{A})\subseteq \overline{f(A)}$ from this question. Can you please ...