-{"_pageContent":{"_pageTitle":"5.1 Continuous Mappings","_hash":"2125c437aaac928d5852224ff00a83f9d9776dda","_attributes":{},"_answers":1},"_parentHash":"6c11e3a31b4b8bd07bdef3f87887ab202a568679","_childPageContents":[{"_pageTitle":"8","_hash":"3e8d61043ed9bdb6b3cd4f197f0203bbdcb13db1","_attributes":{"a.md":{"_time":"2020-10-18T14:48:51+09:00","_attributeFile":{"_content":"I'll refer $`\\mathcal{T}`$ as $`\\mathcal{T}_X`$, $`\\mathcal{T}_1`$ as $`\\mathcal{T}_Y`$, $`\\mathcal{T}_2`$ as $`\\mathcal{T}_A`$, $`\\mathcal{T}_3`$ as $`\\mathcal{T}_B`$.\n\nLet's define $`x \\in X`$ and $`U \\in \\mathcal{T}_Y`$ such that $`f(x) \\in U`$. By definition of continuous mapping, there must exists a $`V`$ such that $`x \\in V \\in \\mathcal{T}_X`$ and $`f(x) \\in fV \\subseteq U`$.\n\nIf for any $`x \\in A`$ and any B-induced open set $`U_B \\in \\mathcal{T}_B`$, which must imply existance of $`U \\in \\mathcal{T}_Y`$, such that $`g(x) \\in U_B`$,\n\n\n\n...there exists an A-induced open set $`V_A`$, which must imply existance of $`V \\in \\mathcal{T}_A`$, such that the image $`gV_A`$ satisfies $`g(x) \\in gV_A \\subseteq U_B`$, then $`g`$ must be continuous. What we have to know is that if the image $`gV_A`$is subset of $`U_B`$, in other words, every $`x \\in V_A`$ will satisfiy $`g(x) \\in U_B`$. Let's proove it.\n\n\n\n```math\n\\begin{aligned}\n & x \\in V_A \\\\\n & \\rightarrow x \\in V \\\\\n & \\rightarrow g(x) \\in gV \\\\\n & \\rightarrow g(x) \\in \\text{ some } U & \\text{ since } f \\text{ is continuous} \\\\\n & \\rightarrow g(x) \\in U \\cap B & \\text{ since codomain of } g \\text{ is } B \\\\\n & \\rightarrow g(x) \\in U_B\n\\end{aligned}\n```"}},"q.md":{"_time":"2020-04-12T01:08:01+09:00","_attributeFile":{"_content":"Let $`(X,\\mathcal{T})`$ and $`(Y,\\mathcal{T}_1)`$ be topological spaces and $`f:(X,\\mathcal{T}) \\rightarrow (Y,\\mathcal{T}_1)`$ a continuous mapping. Let $`A`$ be a subset of $`X`$, $`\\mathcal{T}_2`$ the induced topology on $`A`$, $`B = f(A)`$, $`\\mathcal{T}_3`$ the induced topology on $`B`$ and $`g:(A,\\mathcal{T}_2) \\rightarrow (B,\\mathcal{T}_3)`$ the restriction of $`f`$ to $`A`$. Prove that $`g`$ is continuous."}}},"_answers":1}]}
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