Dense set in a compact subset, then dense in the whole space?

Let $X$ be a compact topological space with continuous map $f\colon X\to X$. $Y:=\bigcap_{n\in\mathbb{N}}f^n(X)\subseteq X$ is closed and hence compact itself.

Assume $\overline{A}=Y$, i.e. $A$ is be dense in $Y$.

Does this imply $\overline{A}=X$, i.e. that $A$ is dense in $X$?

• Did you instead mean $Y \subset \overline{A}$? – B. Mehta May 11 '18 at 13:05
• You can always consider $A=Y\neq X$. So $A$ is trivially dense in $Y$. If $Y$ is closed then $\bar A=Y\neq X$. – user126154 May 11 '18 at 13:24

No it does not. For a counterexample take $Y = [0, 1]$ and $X = [-1, 2]$ as subsets of $\mathbb{R}$ (where $\mathbb{R}$ has the standard topology). Note that $X$ is compact and $Y \subseteq X$ and $Y$ is closed and thus also compact. Then let $A = (0, 1) \subseteq Y \subseteq X$.

Then we have $\overline{A} = Y$ and certainly $\overline{A} = [0, 1] \neq [-1, 2] = X$.

• @Randall Sorry I edited my answer – Perturbative May 11 '18 at 13:16
• Please see the edit of my question. – Rhjg May 11 '18 at 15:35
• @Rhjg You have COMPLETELY changed the question, voiding everyone's work in answering your first question. – Randall May 11 '18 at 17:46
• @Randall if you like you can change the question to its original version again and I will ask this new question in another thread – Rhjg May 11 '18 at 17:54
• @Rhjg Please do so – Perturbative May 12 '18 at 11:22

If $Y$ is a proper subset of $X$, so $X \neq Y$, then $A$ is certainly not dense in $X$. The closure cannot both be $Y$ and $X$ at the same time...

"$A$ dense in $Y$" implies $A \subseteq Y$ and so $\overline{A} \subseteq \overline{Y} = Y \neq X$.

• Please see the edit of my question. I specified Y. – Rhjg May 11 '18 at 15:35
• @Rhjg It doesn't alter my answer. $Y$ is stil closed. What is $A$ precisely, a subset of $Y$? – Henno Brandsma May 11 '18 at 16:40
• Let $x\in X$ be fixed. $A$ is the set of all backward orbits $\{x^{-i}: i\geq 0\}$ where $x^{-i}\in f^{-i}(x)$, $x^0:=x$ and $x^{-i}=f(x^{-(i+1)})$. – Rhjg May 11 '18 at 17:36

Lemma: Given $A\subseteq Y \subseteq X$, where $X$ is a topological space and $Y$ is endowed with the sub-space topology. Then the closure of $A$ in $Y$ is the intersection with $Y$ of the closure of $A$ in $X$: $$\overline{A}^Y=\overline{A}^X\cap Y$$

From this lemma it follows that we can speak about $\overline{A}$ without confusion.

If $Y\neq X$ is closed in $X$, then $\overline A\subseteq Y\neq X$ so no subset of $Y$ can be dense in $X$.

Proof of Lemma: $\overline{A}^X\cap Y$ is closed in $Y$, hence $\overline{A}^Y\subseteq \overline{A}^X\cap Y$ (because $\overline{A}^Y$ is the smallest closed subet of $Y$ containing $A$).

On the other hand, sinche $\overline{A}^Y$ is closed in $Y$, and $Y$ is endowed with the subspace topology, then there is a closed subset $C$ of $X$ such that $\overline{A}^Y=C\cap Y$. As above, since $C$ is closed in $X$, we have $\overline{A}^X\subseteq C$, whence $\overline{A}^X\cap Y\subseteq C\cap Y=\overline{A}^Y$.

Since $\overline{A}^Y\subseteq \overline{A}^X\cap Y\subseteq \overline{A}^Y$, both inclusions are in fact equalities. $\square$
{"ft_lang_label":"__label__en","ft_lang_prob":0.8297517,"math_prob":0.99990666,"size":1084,"snap":"2019-26-2019-30","text_gpt3_token_len":378,"char_repetition_ratio":0.24537037,"word_repetition_ratio":0.03846154,"special_character_ratio":0.30535054,"punctuation_ratio":0.08733624,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.000007,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-06-19T00:48:27Z\",\"WARC-Record-ID\":\"<urn:uuid:f03335af-4ec3-4cfa-bb60-21338fd1130f>\",\"Content-Length\":\"157958\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a507b0b8-049b-47bb-9166-b59b3f47e115>\",\"WARC-Concurrent-To\":\"<urn:uuid:f2d181ee-4223-4e6f-aba5-e46e4ca22a8e>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/2776586/dense-set-in-a-compact-subset-then-dense-in-the-whole-space\",\"WARC-Payload-Digest\":\"sha1:V7UHZUZNXCH2MQWTMGYQAC2KK6M4YX6B\",\"WARC-Block-Digest\":\"sha1:J5E2W56ZJCQ43BH4DHO2G6ERRJ65E3ME\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-26/CC-MAIN-2019-26_segments_1560627998879.63_warc_CC-MAIN-20190619003600-20190619025455-00075.warc.gz\"}"}