Students are given 3 minutes to complete each multiple-choice question on a test and 8 minutes for each free-response question.
There are 15 questions on the test and the students have been given 55 minutes to complete it. A Table titled Test Time, showing Number of Questions, Time per Item in minutes, and Total Time in minutes. The first row shows Multiple Choice, with m, 3, and 3 m. The second row shows Free Response, with 15 minus m, 8, and x. The third row shows Total, with 15, blank, and 55. Which value could replace x in the table? 7 – m 23 – m 8(15 – m) 8(15) – m
1 answer:
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Step-by-step explanation:
If X is a finite Hausdorff space then every two points of X can be separated by open neighborhoods. Say the points of X are . So there are disjoint open neighborhoods
and
, of
and
respectively (that's the definition of Hausdorff space). There are also open disjoint neighborhoods
and
of
and
respectively, and disjoint open neighborhoods
and
of
and
, and so on, all the way to disjoint open neighborhoods
, and
of
and
respectively. So
has every element of
in it, except for
. Since
is union of open sets, it is open, and so
, which is the singleton
, is closed. Therefore every singleton is closed.
Now, remember finite union of closed sets is closed, so is closed, and so its complemented, which is
is open. Therefore every singleton is also open.
That means any two points of belong to different connected components (since we can express X as the union of the open sets
, so that
is in a different connected component than
, and same could be done with any
), and so each point is in its own connected component. And so the space is totally disconnected.