I was wondering if one can construct a non-continuous function $f:X\to Y$ between two topological spaces $X$ and $Y$, such that $f$ sends every convergent net into convergent net... Thanks in advance.

Let $f:(X,\mathcal T)\to (Y,\mathcal S)$ be a function between topological spaces. Let for any convergent net $(x_\alpha)$ in $X$, $(f(x_\alpha ))$ be convergent in $Y$. Is $f$ continuous? (It seems to be true in completely regular spaces).

Let $\{ E_n \}_{n \in \mathbb{N} }$ be a sequence of sets in some ambient set $\Omega $. I want to show that $$ \liminf E_n \subset \limsup E_n $$ My attempt: IF $x \in \liminf E_n = \bigcup_{k=1}^{\infty} \bigcap_{n \geq k} E_n $, then there is some $k_0 \in \mathbb{N}$ so that $x \in \bigcap...

I am wondering if my proof is correct? Thank you for whoever willing to take a look at it for me. Proof $\liminf E_k \subset \limsup E_k $ If $\{E_k\}_{k=1}^\infty$ is a sequence of sets, we define \begin{align*} \limsup E_k & = \bigcap_{j=1}^\infty\left(\bigcup_{k=j}^\infty E_k\right)\\ &=...