# Subsets of not connected topology

I was taking a look at this book. It says (p 102) that

If a topological space (X, T) is not connected, than X contains a subset U such that U isn't X, U isn't empty and U is clopen.

The book brings no proof of the fact. How does this imply?

-

## 2 Answers

This follows immediately from the fact that if $X$ is not connected, we can write

$$X = U \cup V$$

with $U$ and $V$ disjoint, not empty and both open in $X$. Since they are disjoint and their union is the whole space, we have $U = X- V$ and so $U$ is closed at the same time.

-

The answer depends on which definition of connectedness is being used. In this case it’s trivial, because the definition used in this book is as follows:

3.3.4 Definition. Let $(X,\tau)$ be a topological space. Then it is said to be $\color{red}{\text{connected}}$ if the only clopen subsets of $X$ are $X$ and $\varnothing$.

In other words, $X$ is not connected if and only if it contains a clopen set other than $X$ and $\varnothing$.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 14, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9750088453292847, "perplexity_flag": "head"}