# Linked Questions

5answers
5k views

1answer
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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\...
8answers
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3answers
209 views

### 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 $(\mathbb{R},T_{cof})$, ...
1answer
736 views

### 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 ...
2answers
331 views

### 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 $x ∈ R$.
4answers
114 views

### Example of a continuous function s.t. $f(\overline{A}) \subsetneq \overline{f(A)}$

This question is a subproblem of the A map is continuous if and only if for every set, the image of closure is contained in the closure of image. I am not able to come up with any example of a ...
3answers
201 views

### 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 ...

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