Welcome to mathimatikoi.org forum; Enjoy your visit here.

## Semicontinuity

General Topology
Tsakanikas Nickos
Community Team Articles: 0
Posts: 314
Joined: Tue Nov 10, 2015 8:25 pm

### Semicontinuity

Definition: Let $Y$ be a topological space. A function $\varphi \ \colon Y \longrightarrow \mathbb{Z}$ is called upper semicontinuous if for every $y \in Y$ there exists an open neighborhood $U$ of $y$ such that $\varphi(y) \geq \varphi(y^{\prime})$ for all $y^{\prime} \in U$.

Show that a function $\varphi \ \colon Y \longrightarrow \mathbb{Z}$ is upper semicontinuous if and only if for every $n \in \mathbb{Z}$ the set $\left\{ \, y \in Y \ \big| \ \varphi(y) \geq n \, \right\}$ is a closed subset of $Y$.
{"ft_lang_label":"__label__en","ft_lang_prob":0.63332266,"math_prob":0.99807256,"size":676,"snap":"2021-21-2021-25","text_gpt3_token_len":233,"char_repetition_ratio":0.11160714,"word_repetition_ratio":0.09917355,"special_character_ratio":0.3402367,"punctuation_ratio":0.11111111,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9981381,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-18T15:11:15Z\",\"WARC-Record-ID\":\"<urn:uuid:de48775b-9db1-4ba2-9ec0-1bffeec7aaf1>\",\"Content-Length\":\"34000\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:eea9e369-a684-460c-99bc-51522d71a08a>\",\"WARC-Concurrent-To\":\"<urn:uuid:d14a287d-bdee-4841-87a8-724d22d3761b>\",\"WARC-IP-Address\":\"213.108.198.204\",\"WARC-Target-URI\":\"https://mathimatikoi.org/forum/viewtopic.php?f=15&t=991&p=2051&sid=63c03d658c3d34f21b54a24a0e67f6b1\",\"WARC-Payload-Digest\":\"sha1:GPXWAV67A2O55T4V2HWQHQPCNP2USJKG\",\"WARC-Block-Digest\":\"sha1:6YDP3AWINGVMEE4TTWN4YEEEW7UZQH76\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623487637721.34_warc_CC-MAIN-20210618134943-20210618164943-00530.warc.gz\"}"}